
P. GonzálezMazón, A. Hubard, R. Karasev. “Weak (1ε)nets for polynomial superlevel sets”, 2023.
The classical centerpoint theorem, given a measure in the Euclidean space, provides a point such that any halfspace containing that point also contains a certain fraction (depending on the dimension) of the measure.
We replace halfspaces by sets defined by a polynomial inequality of given degree and observe that one centerpoint is not enough then. We have to take a finite set of points so that any given degree polynomial superlevel set containing all those points also contains a certain fraction (depending on the dimension and the degree) of the measure.
Trying to bound the size of the centerpoint set from above we in fact have to address a geometric problem, the Carathéodory number of the Veronese map image. This question and its modifications have been studied to some extent. Here we list the known results, provide some new bounds, and indicate wide open questions.

M. Berezovik, R. Karasev. “Symplectic polarity and Mahler‘s conjecture”, 2022.
In this paper we consider symplectic version of the polarity transform and symplectically selfpolar convex bodies. Unlike the Euclidean case, there exist various symplectically selfpolar bodies.
We prove the simple sharp upper bound on the volumes of such bodies and show that the conjectured lower bound is equivalent to Mahler‘s conjecture about volumes of a centrally symmetric convex body and its Euclidean polar. We establish lower and upper bounds on symplectic capacities of symplectically selfpolar bodies that directly confirm consistency of the volume lower bound with Viterbo‘s conjecture on the volume and the capacity. Besides that, we make other relevant observations around the notion of symplectic polarity.

A. Glazyrin, R. Karasev, A. Polyanskii. “Extensions of polynomial plank covering theorems”, 2022.
We continue our work arXiv:2112.05382 and prove the complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round.
We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths. The version of this conjecture with sharp inequality remains unsettled.

A. Kamal, R. Karasev. “Systolic inequalities and chromatic number”, 2022.
We show that the discrete versions of the systolic inequality (originated in the work of Gromov in Riemannian geometry) that estimate the number of vertices of a simplicial complex from below have substantial applications to graphs, the onedimensional simplicial complexes. Almost directly the discrete systolic inequality provides good estimates for the number of vertices of a graph in terms of its chromatic number and the length of the smallest odd cycle.
Combined with the graphtheoretic techniques of Berlov and Bogdanov that were previously applied to the same problem, the argument used in the systolic proof produces even better estimates for the number of vertices.

R. Karasev, A. Sharipova. “Convex bodies with all characteristics planar”, 2022.
We show that smooth and strongly convex bodies in the symplectic R^{2n} for n>1 with all characteristics planar (contained in affine twodimensional planes), or all outer billiard trajectories planar are linear symplectic images of balls. This is a modification of the natural conjecture that smooth convex bodies having all characteristics closed with the same action are symplectomorphic images of balls.

We note that the recent polynomial proofs of (particular cases of) the spherical and complex plank covering problems by Zhao and OrtegaMoreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. After that we establish polynomial analogs of the Bang theorem by explaining how to find a point in the unit ball sufficiently far from the zero set of a given polynomial.
As a corollary of these results, we establish a conjecture of Jiang and Polyanskii about covering a sphere by spherical segments generalizing the zone conjecture of Fejes Tóth.

In the previous two papers we studied compexity of triangulations in terms of the number of vertices. This time we study lower bounds on the number of
n dimensional faces. In the presence of the fundamental group and nontriviality of ann dimensional group cohomology class of the space, we prove the lower bound of2^{n} . This generalizes the previously known estimates for triangulations of RP^{n} by Bárány and Lovász.This estimate does not seem very strong in general, but our argument applies not only to simplicial complexes, but also to more general objects, regular simplicial sets. Informally, those are the spaces obtained by gluing simplices together so that every simplex embeds homeomorphically. Viewed this way, our estimate is sharp for the projective space and provides a nontrivial (but most probably not sharp) bound for the
n dimensional torus. 
In this paper we present a combinatorial analogue of Gromov‘s systolic inequality. This analogue provides a lower bound on the number of vertices of a simplicial complex in terms of its edgepath systole, the shortest noncontractible loop along its edges.
Similarly to the Riemannian case, where the inequality holds under a topological assumption of “essentiality”, our proofs rely on a combinatorial analogue of essentiality. Under a stronger assumption, expressed in terms of cohomology cuplength of the simplicial complex, we prove quantitatively stronger estimates.
We also illustrate our methods (an evolutionary development of the known methods) in the continuous (piecewise Riemannian) setting, generalizing and improving quantitatively the “Minkowski principle for cohomology products” of Balacheff and Karam. A corollary of this result is the extension of the GuthNakamura cuplength systolic bound from manifolds to piecewise Riemannian spaces.

S. Avvakumov, R. Karasev. “Equipartition of a segment”, 2020.
We prove that, for any positive integer m, a segment may be partitioned into m possibly degenerate or empty segments with equal values of a continuous function f of a segment, assuming that f may take positive and negative values, but its value on degenerate or empty segments is zero.

K. Adiprasito, S. Avvakumov, R. Karasev. “A subexponential size triangulation of RP^{n}”, 2020.
In this paper my colleagues, with a little help from me, construct a polytope with subexponential number of vertices in terms of dimension and whose quotient is a triangulation of the real projective space.

R. Karasev, A. Skopenkov. “Some ‘converses’ to intrinsic linking theorems”, 2020.
A lowdimensional version of our main result is the following ‘converse’ of the ConwayGordonSachs theorem on intrinsic linking of the graph K_{6} in 3space: For any odd integer m, there is an embedding of the graph K_{6} in space such that the linking number of the images of triangles 123 and 456 of this graph is m, while the linking number of images of any other pair of disjoint triangles in this graph is zero.
We also prove a higherdimensional analogue, which is in the metastable range a ‘converse’ of a lemma by SegalSpież. Above the metastable range we have a weaker result about almost embeddings instead of embeddings, which leaves open questions.

A. Akopyan, R. Karasev. “When different norms lead to same billiard trajectories?”, 2019.
We follow the works of Milena Radnovi? and Serge Tabachnikov, studying the questions when two different not necessarily symmetric norms produce the same billiard reflection law and the same billiard trajectories.
It is clear from the symplectic point of view on billiards that whenever we translate or positively scale the polar of the unit ball of the norm, the billiard trajectories do not change. It remains to note that the original unit ball is then changed by a projective transformation that fixes the origin and preserves every line through the origin without changing its orientation at the origin.
Our observation is connected to the previous work by the fact that a Euclidean ball centered at the origin under such a projective transformation becomes an ellipsoid of revolution about a line that has a focus at the origin.

We show that the deleted product obstruction for removing rtuple points of maps to the Euclidean space of dimension at least 2 totally vanishes if r is not a prime power.
As a consequence of this, we improve the existence results of such maps without rtuple points from pairwise disjoint faces and build stronger counterexamples to the topological Tverberg conjecture for nonprimepower r.

R. Karasev. “Huang's theorem and the exterior algebra”, 2019.
In this note we give a version of Hao Huang's proof of the sensitivity conjecture, shedding some light on the origin of the magical matrix A in that proof. This text only contains the proof of the core fact: Consider the ndimensional boolean cube Q_{n} as a graph, whose edges connect pairs of vertices differing in one coordinate. Then any its induced subgraph on greater than 2^{n1} (the half) vertices has degree of some vertex at least square root of n.

S. Avvakumov, R. Karasev. “Envyfree division using mapping degree”, 2019.
In this paper we study socalled envyfree division problems. The classical approach to some of such problem reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the KnasterKuratowskiMazurkiewicz theorem) is a boundary condition.
This time we try to replace the boundary condition by a certain equivariance condition under all permutations, or a weaker condition of “pseudoequivariance”, which has some economic meaning for the problem of partitioning a segment. Such versions of the problem have positive solutions when then number of persons among whom we partition is a prime power, and in some cases we provide respective counterexamples for the case when the number of persons is not a prime power.

R. Karasev. “Mahler's conjecture for some hyperplane sections”, 2019.
In this paper I use the symplectic interpretation of Mahler's conjecture to establish its particular cases, for hyperplane section of l_p balls and Hanner polytopes, and for their hyperplane projections. Those particular cases (and conjecturally the general case) are related to the study of the behavior of the symplectic volume under the symplectic reduction.

We study some particular cases of Viterbo's conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately. We manage to establish the conjecture for sublevel sets of some convex 2homogeneous Hamiltonians of this kind and in one more relatively easy case: we also discuss open particular cases of this conjecture.

This paper continues the study of bisection of measures by several hyperplanes in the Euclidean space, started in arXiv:1803.02842. A measure is bisected by an arrangement of D hyperplanes if, after properly coloring the cells of the complement of the arrangement of hyperplanes in two colors, the measure of each color is the half of the whole measure.
Apart from the previously established case when the dimension n is a power of two, we establish one more similar case with slightly nonoptimal number of hyperplanes. We also note that the case D=2 of this problem is equivalent to the projective ham sandwich theorem, previously established by Pavle Blagojevic and me in [arXiv:1011.0869, Theorem 10].

In this paper we consider the Hedetniemi conjecture about the chromatic number of the product of graphs, and its generalization by Zhu for hypergraphs.
We verify this conjecture to certain families of graphs, and we also study the closely related conjectures about the equivariant index of the product of two Z/pspaces, where p is a prime. For p=2, when the geometrical index is replaced by the homological index, we confirm the conjectured value of the index of the product.

In this paper we continue to study the Gromov waist in the sense of tneighborhoods for measures in the Euclidean space. Unlike our recent papers, now we are interested in an estimate holding for arbitrary value of t, not only asymptotically for t tending to 0. We are motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures and try to extend this results to other measures. In particular, it turns our possible to have a sharp estimate in the case of not necessarily radially symmetric Gaussian measure.
We also provide examples of measures having no tneighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov's pancake argument to produce some estimates of tneighborhoods of (weighted) volumecritical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader's convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures.

V. Bui, R. Karasev. “On the Caratheodory number for strong convexity”, 2018.
In this paper we once again touch the topic of Kstrong convexity, where K is a given convex body and the Kstrong convex hull of a point set X is the intersection of translates of K that contain X.
We investigate the Caratheodory number for Kstrong convexity and prove some general facts about it. In particular, we show that this number is always greater or equal to the dimension of K. This bound is attained for nontrivial products of “generating sets”, in particular for products of convex bodies of dimension at most 2. We also give a simpler example of K with unbounded Caratheodory number and make several other observations on the topic.

In this paper we continue to investigate the problem (by Nandakumar and Ramana Rao) of cutting a convex shape in the plane into m convex parts of equal areas and perimeters. In the series of previous works the existence problem was solved affirmatively for m=p^{k}, a prime power. This paper (in its second version) establishes the result for arbitrary number of parts m.

A. Hubard, R. Karasev. “Bisecting measures by hyperplane arrangements”, 2018.
In this paper we study the following problem: How many measures in R^{n} can be simultaneously bisected by D hyperplanes? A measure is bisected by an arrangement of D hyperplanes if, after properly coloring the cells of the complement of the arrangement of hyperplanes in two colors, the measure of each color is the half of the whole measure.
We show that when the dimension n is a power of two, a set of nD measures can be bisected simultaneously; this is tight by the dimension considerations. For other values of n we only have partial result be reducing to the case of power of two.

In this paper we improve the estimate of Gluskin and Ostrover for the EkelandHoferZehnder capacity of a convex body in the symplectic R^{2n}. The estimate follows from a lower bound of the length of a closed characteristic on the boundary of K in the norm (gauge) with unit ball K, which is similar to what we studied in 1401.0442.
In the case of centrally symmetric K we show that one of the closed characteristics of the minimal action on the boundary of K must itself be centrally symmetric. This allows to use the length lower bounds established by Schaffer. We generalize this symmetry observation to the case when K is invariant with respect to multiplication by a root of unity, under the identification R^{2n}=C^{n}.

In this paper we continue the work of arXiv:1006.2263 and study the involution on the real oriented or nonoriented Grassmannian of halfdimensional subspaces, sending a subspace to its orthogonal complement. This time we manage to calculate the homological index of this action for all dimensions, which allows to infer certain BorsukUlamtype theorems.

A.V. Akopyan, R.N. Karasev. “Waist of balls in hyperbolic and spherical spaces”, 2017.
In this paper we study the waist theorems in the sense of Gromov for balls in the model spaces of constant curvature. In particular, we prove that for a continuous map from such an ndimensional ball to the Euclidean space of dimension k, the (nk)dimensional Minkowski content of some fiber of the map is at least the volume of the (nk)dimensional ball of the same radius and same curvature.
We also obtain some results for the balls in the spaces of bounded from above curvature (CATspaces) and not necessarily symmetric normed spaces. Along the way we study the behavior of the Minkowski content under Lipschitz maps between Riemannian manifolds.

We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous question in the field is the “weak simplex conjecture”, saying that the regular simplex is optimal among other sets of n+1 points on the unit (n1)dimensional sphere.
We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the Šidák inequality and establish some theoretical and some numerical results about their optimality. 
We continue investigating Gromov's waist of different spaces. Several results of the paper follow the idea of Bo'az Klartag, also used by two of us in the work arxiv:1608.06279.
We also extend some of the results from waists in terms of fibers of a continuous map to the waist in terms of families of cycles carrying a nonvanishing power of the fundamental cohomology class.
The third direction is to address the waist in terms of the Hausdorff measure of fibers for maps into arbitrary polyhedron, with a constant given in terms of the dimension only.

Dolnikov, Zivaljevic, and Vrecica established the center transversal theorem: Any m nice measures in R^{n+m1} can be linearly projected to R^{n} so that they have a common point with depth at least 1/(n+1) with respect to any of the projected measures.
Following the previous result of Magazinov and Por, we slightly improve the constant 1/(n+1) in expense of increasing the dimension from n+m1 to 2m+n1 or 3m+n1 depending on whether n+1 is a power of two or not.

A.V. Akopyan, R.N. Karasev. “A tight estimate for the waist of the ball”, 2016.
We use the idea of Bo'az Klartag to prove Gromovtype waist theorems for the Minkowski measure using a Lipschitz map of one measure to the other; and answer a question of Gromov about the waist of the Euclidean ball.
Another tool of the proof in the fact, going back to Archimedes, that the standard projection from the (n+1)dimensional sphere to the ndimensional ball maps the uniform measure in the sphere to a multiple of the uniform measure in the ball.

P.V.M. Blagojevic, R.N. Karasev. “Local multiplicity of continuous maps between manifolds”, 2016.
This paper develops the ideas from the previous preprints arxiv:1002.0660 and arxiv:1106.6176 about the ways to guarantee the multiplicity of a continuous map between two manifolds.
This time we have done more careful calculation using the previous work of Pavle and managed to write down the characteristic classes responsible for local multiple points.
Warning. We have noticed an error in the calculation and will try to correct it, possibly arriving at a different final formula for the characteristic classes.

A. Balitskiy, R. Karasev. “Dependence of the heavily covered point on parameters”, 2016.
The "first selection lemma" by Imre Bárány asserts that given many points in R^{n} it is possible to find another point p such that the probability that it is covered by a randomly chosen simplex formed by some n+1 of the given points is at least some positive constant c_{n}.
Recently Gromov designed a new technique to prove such results, improving the constant c_{n}. We examine Gromov’s method in order to understand the dependence of such a "heavily covered" point on parameters. We cannot prove the continuous dependence on parameters, but manage to utilize the "homological continuous dependence" of the heavily covered point. This allows us to infer some corollaries in a usual way.
We also give an elementary argument to prove the simplest of these corollaries: Given several straight lines in the plane in general position, it is possible to find a point that is surrounded by a randomly chosen triple of lines with probability at least 2/9.

A. F. Holmsen, R. N. Karasev. “Colorful theorems for strong convexity”, 2015.
In this note we discuss certain analogues of the Carathéodory theorem for the notion of strong convexity. In my PhD and a paper published in Russian I established such a theorem in 2001. Now we generalize that result to have an analogue of the colorful Caratéodory theorem, established for ordinary convexity by Imre Bárány, and the “very colorful theorems”, established by several people including Andreas Holmsen.
These theorems require the gauge body to satisfy a certain “generating convex set” assumption introduced by Polovinkin and Balashov. This assumption is satisfied by any convex body in the plane, but in 3space we give an example where this assumption fails and the corresponding analogue of the Carathéodory theorem also fails.
We also try to give a topological criterion for one convex body to be a Minkowski summand of another. We managed to do so, but under a somewhat artificial assumption that one of the bodies is closed and the other one is open.

János Pach proved that there exists a constant c_{d} such that for any collection of finite sets X_{0}, ... , X_{d} in R^{d} it is possible to choose the subsets Y_{i} of X_{i} for every i and a point p in R^{d} so that Y_{i} is no less than c_{d}X_{i} and every system of representatives of the collection Y_{0}, ... , Y_{d} contains point p in its convex hull.
This paper provides upper bounds for the constant c_{d} by providing examples of the collection of finite sets. Actually, the example is rather straightforward, while the calculation of the constant is not so. In particular, there arises the problem of bounding from above the minimal solid angle at a vertex of arbitrary simplex in R^{d}. This is where my contribution allowed to improve the bound on the solid angle, given in Theorem 6 of the current version of the text.
The problem of bounding the minimal solid angle of a simplex is also studied in the parallel paper of Arseniy Akopyan and myself, where precise estimates in dimension d=3 and 4 are given. This paper is described below.
Also available on arxiv.org.

A.V. Akopyan, R.N. Karasev. “Bounding minimal solid angles of polytopes”, 2015.
We investigate the following question raised by Kynčl, Paták, Safernová, and Tancer: Does every simplex in R^{n} has a solid angle at a vertex that is no greater than the solid angle at any vertex of a regular simplex in R^{n}?
It turned out that it is possible to answer the question affirmatively for dimensions 3 and 4. This argument uses certain facts about the isoperimetry for simplices in spherical geometry. We also formulate several generalizations for other regular polytopes.
Update after peer reviewing: In turned out that we misunderstood the result on isoperimetry for spherical tetrahedra in the paper of Böhm. So the case of dimension 4 in our problem remains open.

R.N. Karasev. “Residues and the Combinatorial Nullstellensatz”, 2015.
We start from the expression of the Combinatorial Nullstellensatz of Noga Alon as a certain equality, connecting a coefficient of a multivariate polynomial and its values at certain points. This formula was proposed by Fedya Petrov in our joint paper “Partitions of nonzero elements of a finite field into pairs”.
In this text we observe that this formula is actually a multidimensional residue formula. This point of view allows to better understand old and prove some new results. We also mention some open (or partially open) problems about point configurations with algebraic flavor.

This paper was written by my coauthors Pablo and Edgardo, who studied the question of simultaneous partitioning several measures into two (or more) equal parts in a way, similar to the “ham sandwich” and “splitting necklace” theorems. The partitions were made by iterated hypeplane cuts, where the directions of the hyperplanes were prescribed.
My contribution is explained in Section 4, where we consider certain very general Voronoi partitions and make a statement that generalizes Noga Alon's “splitting necklace” theorem and Rade Živaljević's “curtain partition” theorem at the same time. Also, with a small technical effort this approach handles iterated partitions as well.

A.V. Akopyan, R.N. Karasev, F.V. Petrov. “Bang's problem and symplectic invariants”, 2014.
In this paper we find more applications of the symplectic techniques to convex geometry. We address the TarskiBang problem about covering a convex body by “planks” (regions between two parallel hyperplanes) and its relatives; we examine what can and what cannot be done using the symplectic techniques.
The translation of the TarskiBang problem to the symplectic setting, along with the results of Keith Ball, gives rise to a certain conjecture about the subadditivity property of symplectic capacities. We are not able to establish this property fully, but we prove some its particular cases. We also prove several related results with nonsymplectic techniques, including one new particular case of the TarskiBang problem in its original form.

We continue to study billiards in convex bodies, presumably related to the Mahler conjecture on the product of volumes of a convex body and its polar. This time we consider norms that are not symmetric with respect to the origin, one may call this case flat Finsler billiard.
Our goal is to make lower bounds on the length of a closed billiard trajectory in a convex body. The technique of Karoly Bezdek and Daniel Bezdek, developed in the Euclidean case, works in this asymmetric norm case as well. This allows to give elementary proofs of some known results (previously proved with symplectic techniques) and prove the tight estimate for the shortest closed billiard trajectory in a convex body measured with the norm taking this convex body as a unit ball.
The third version of the paper has changed its title. The previous title was “Elementary results in nonreflexive Finsler billiards”.

In this text we consider geometric joins of a family of m finite point sets X_{1}, X_{2}, ..., X_{m} in ddimensional Euclidean space. This is just the set of all possible convex combinations of systems of representatives of this family.
It is not hard to prove that for sufficiently large m this set becomes contractible, and we prove this for m>d(d+1)/2, and also prove that for m>d(d+1) it is startshaped. But we conjecture that a geometric join of m sets is contractible already for m=d+1, and we manage to prove this conjecture for d = 2 and d = 3.
Here is an arxiv link to this paper.

Rade Živaljević proved a version of ham sandwich theorem in arbitrary dimension, named &lsqou;the curtain partition theorem’. Its 2dimensional case is rather simple: Given two nice measures in the plane and a 3fan, one can choose an angle of the 3fan and translate it so that the resulting angle cuts precisely half of every measure.
We extend this result in the plane by considering fans with arbitrary number of angles. In some cases it is possible to prove the positive assertion, and in some cases we find counterexamples.
Here is a link to the arxiv version of this paper.

In this paper I observe that the "quantitative covering dimension" theorems, like the Lebesgue theorem about the cube and the KnasterKuratowskiMazurkiewicz theorem about the simplex, have a simple explanation in terms of toric geometry.
This provides a unified point of view on such results and allows to generalize them to some extent. In particular, it gives an answer to a question of Domotor Palvolgyi on mathoverflow.net.
Available at arxiv.org.

This paper is merged from the two texts: arXiv:1011.4762 of myself and arXiv:1010.4611 of Alfredo Hubard and Boris Aronov.
In this version we make a clearer and slightly more general statements of the main theorems and spend some effort to explain the proof of the topological lemma. Our approach to the topological lemma is the same that D.B. Fuks and V.A. Vasiliev used to establish its particular cases. Another, more combinatorial approach to the topological lemma can be found in the paper arXiv:1202.5504 of P. Blagojevic and G. Ziegler.
Available at arxiv.org.

In this paper we present a new approach to the Mahler conjecture relating it to the Viterbo conjecture from symplectic geometry.
My coauthors have already established the correspondence between the shortest closed billiard length in a convex body K measured with a norm and the HoferZehnder symplectic capacity c(KxT), where T is the unit ball of the dual norm. In this paper we establish the following lower bound by nonsymplectic methods: this value is at least 4 when K is the unit ball of a norm and T is its polar body (unit ball of the dual norm).
In symplectic geometry, Viterbo conjectured that the volume of a convex body X in R^{2n} is at least c(X)^{n}/n!. If it is true then vol KxT ≥ 4^{n}/n! for a centrally symmetric convex body in R^{n} and its polar, this is the classical Mahler conjecture. So to finish the Mahler conjecture it remains to prove the Viterbo conjecture.
Available at arxiv.org.

B. Knaster conjectured that for a continuous function f on the round sphere of dimension n1 and a finite subset X of n points on the sphere it is always possible to find an isometric copy X' of X on the sphere so that f becomes constant on X'. This conjecture turned out to be wrong, as proved by Boris Kashin and Stanislaw Szarek, though some particular cases were established.
We analyze the known approaches to particular cases of Knaster's problem and notice that in many cases the set X is required to be an orbit of a finite group action. Morevover, this group G is usually a ptorus, that is a product of several copies of the same cyclic group of prime order. We then observe that a weak form of the Knaster conjecture (when the dimension of the sphere is allowed to be arbitrarily larger than the cardinality of X) holds for sets X that can be included into an orbit of a ptorus, we call them spherical subptoral sets.
In the Euclidean Ramsey theory, that studies monochromatic sets in an arbitrary coloring of the Euclidean space in a given number of colors, the notion of a suborbit of a finite group action already proved to be useful. By borrowing ideas from Ramsey theory we prove weak Knaster properties of nonequatorial triangles in spheres, and of simplices in Euclidean spaces. We also find some sets already on S^{1} that are not subptoral for any p and show that Knastertype approaches to the Dvoretzky theorem are still insufficient.
Available at arxiv.org.

In this paper we make the following observation about the topology of the Stiefel manifold of orthonormal kframes in R^{n}: The Schwarz genus of the natural action of W_{k} on this space is maximal possible, that is equal to its dimension plus one. Here W_{k} is the group that permutes the frame vectors and changes signs of some of them.
This result has certain geometric consequences about critically outscribed parallelotopes for a given convex body and Birkhoff–James orthogonal bases in normed spaces, see the definitions in the text. In both cases we estimate the number of such objects from below by n( n 1)/2+1, where n is the dimension.
This paper is also available on arxiv.org.

A.V. Akopyan, R.N. Karasev, A.Yu. Volovikov. “BorsukUlam type theorems for metric spaces”, 2012.
In this paper we start from M. Gromov's method of contraction in the space of (co)cycles, and observe that this method applies well to the classical theorems of BorsukUlam and Hopf. This method turns out to be useful and allows us to prove certain generalizations of these theorems.
Using those topological results, we study some analogues of the notion of the Urysohn width and the Gromov waist for metric spaces, and prove some results in this direction. However, in the field of widths and waists there remain a lot of open questions.

The center point theorem asserts that for a finite point set X in R^{d} one may select the center point c so that every halfspace containing c must contain at least 1/(d+1) of the points of X.
In one of my previous papers the "dual" center point theorem was proved: For a finite set X of hyperplanes in general position one may select the center point c so that every ray starting at c intersects at least 1/(d+1) members of X. This theorem does not follow from projectively dualizing the original center point theorem. Its projective dualization gives another fact, I let the reader check it.
Benjamin Matschke proposed to make a theorem that interpolates between the original center point theorem and its "dual" version. The main idea is to start from a subspace V of RP^{d} and find another subspace W of complementary dimension ddimV1 so that every pair of hyperplanes H and J containing V and W respectively does not cut too little of the given finite point set X. Here it is miportant that a pair of hyperplanes partition the projective space in two parts.
So we prove a general projective center point theorem as well as the corresponding version of the Tverberg theorem. Then we give a very general theorem that also includes the Tverberg transversal theorem as a particular case, and stude several similar problems.
Available at arxiv.org.

We study the possibility to cut the same fraction of d+1 measures in R^{d}.
One particular case is cutting by a hyperplane a same (not prescribed) fraction of every measure. In general it is not possible, but under some additional assumptions in term of geometric permutations this is possible.
The other case is to cut a prescribed fraction α of every measure with a convex set. We show that (without other assumptions) this is possible for α=1/m with positive integer m and is not possible for other α.
Some remaining open problems are also outlined in this paper.
This paper is also available on arxiv.org.

The Carathéodory theorem asserts that a point x belongs to the convex hull of a subset X in R^{n} if and only if it belongs to the convex hull of some at most n+1 element subset of X.
Several cases were known when the constant n+1 can be made less. For example, the Fenchel theorem asserts that this constant is at most n for a compact X whose components cannot be separated by a hypeplane.
In this note we consider results resembling that theorem of Fenchel. We also prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg type theorem for families of convex compacta.
This paper is also available on arxiv.org.

We consider a ddimensional cube Q subdivided into n^{d} small cubes in an obvious way. Then we color the small cubes into m+1 colors and want to have some lower bound on the size of a monochromatic connected component.
For m=d1 (which is d colors) it is well known that one monochromatic connected component connects two opposite facets of Q thus giving the lower bound n. For m=1 (2 colors) Matousek and Privetivy established the lower bound approximately n^{d1}.
In this paper Gromov's method of contraction in the space of cycles allows to prove the lower bound f(d,m) n^{dm} with some nonoptimal coefficient f(d, m).
This paper is also available at arxiv.org.

The Turan number ex(H, n) of a graph H is the maximum number of edges in a graph G on n vertices having no subgraph isomorphic to H. For full bipartite graphs H=K_{s,t} there exist constructions of G with large number of edges given by an algebraic relation between two points in F_{p}^{d}.
In this paper we study such relations on R^{d} or C^{d} that are defined over integers and therefore give graphs over F_{p}, for which asymptotic (as p grows) number of edges can be found.
We show that any hypersurface in R^{d}xR^{d} must contain a grid (which is a product of finite sets SxT) of size approximately (2d3)x(2d3) by a topological reason. This rules out the construction of corresponding K_{s,s}free graphs (interesting from the point of view of the Turan problem) with s at least 4. On the other hand we construct hypersurfaces in R^{d}xR^{d} with no dxd^{4d}grid. The construction of hypersurfaces without dxd^{2} grids, for example, remains an open question.
This paper is also available on arxiv.org.

In this paper we consider the following question: Does every convex polytope P admits an inscribed regular octahedron? By “inscribed” we mean that the vertices of the octahedron are on the boundary of P.
For smooth convex bodies in place of P, and more generally, for smooth embedded 2spheres in space the answer was already shown to be positive by Vladimir Makeev. But the usual “going to the limit” argument does not allow to deduce the result for a plytope from the result for a smooth surface.
In this paper we make going to the limit carefully enough to answer the question positively for simple polytopes P. The case of nonsimple P generally remains open.
This paper is also available at arxiv.org

R.N. Karasev. “Geometric coincidence results from multiplicity of continuous maps”, 2011.
A couple of typical geometric problems about coincidence were posed by B. Grünbaum: How many affine diameters of a convex body in R^{n} must have a common point? How many centers (in some reasonable sense) of hyperplane sections of a convex body in R^{n} must coincide?
One possible approach to such problems is to find topological reasons for multiple coincidences for a continuous map between manifolds of equal dimension. In other words, we need topological estimates for the multiplicity of a map. In these particular problems we have to estimate from below the multiplicity of any continuous map from the projective space to the sphere of the same dimension. For certain values of n some nontrivial lower bounds are obtained; though they are far from n+1, which was conjectured by Grünbaum.

For convex partitions of a convex body B we try to put a homothetic copy of B into each set of the partition so that the sum of homothety coefficients is at least 1. In the plane it is possible for any partition, while in higher dimensions we need certain restrictions on the partition.
This problem has much in common with the Bang conjecture. The difference is that in the Bang conjecture instead of a partition of B into convex parts we consider a covering of B by rather simple objects: planks between pairs of parallel hyperplanes. Even in the plane these two problems are not equivalent; so we make no advance in the Bang conjecture.
You may also download this paper from arxiv.org.

In this paper we study higher topological complexity of spheres. Less formally, we study the following question: Can one span any mtuple of distinct points in the sphere S^{n} by a continous image of the wedge of m segments so that everything depends continuously on the mtuple.
This question for m=2 was studied before as a natural question of planning the motion between two points on the sphere.
We prove that the answer to this question in negative in most cases. The proof is based on examining the maps between cohomology of certain spaces and the generalized Hopf invariant of maps. In fact, we do not know any construction giving a positive answer to this question. The smallest remaining open case is m=3 and n=4.
If you do not have access to the journal version, you can download the paper from arxiv.org.

R.N. Karasev and A.Yu. Volovikov. “Waist of the sphere for maps to manifolds”, 2011.
In this paper we generalize the sphere waist theorem of Gromov and the BorsukUlam type measure partition lemma of GromovMemarian for maps to manifolds.
Informally, this result claims that for any continuous map f : S^{n}> M to a manifold of dimension m < n the preimage of some point is at least as "large" as the standard (nm)dimensional subsphere of S^{n}.

In this note the following result is proved: Suppose d+1 vertices are distributed in R^{d} randomly and independently; then there exists a "central point" c such that the probability for this point to be covered by the random siimplex is at least p_{d}=1/(d+1)!.
In case of discrete and equal distributions for all vertices this theorem was proved by Imre Barany with estimate on p_{d} of order 1/(d+1)^{d+1}. That proof used the Tverberg theorem and the colorful Caratheodory theorem. Boros and Furedi improved the esitmate for d=2 to 2/9, and Janos Pach considered a similar problem for d+1 vertices with different distributions.
In 2010 Mikhail Gromov published another approach to this problem and obtained better bounds (the same as in this note). But the proofs were not elementary and not accessible to most experts in discrete geometry.
In this note the proof of Gromov is translated to the elementary language, its size dramatically shrunk. Another difference in this note is considering different distributions for different vertices.
If you do not have access to SpringerLink, you can download the paper from arxiv.org.

R.N. Karasev. “Equipartition of several measures”, 2010.
In this paper I prove several results of the following type: any d measures in R^{d} can be partitioned simultaneously into k equal parts by a convex partition (this particular result was proved before by Pablo Soberon). Another example is: any convex body in the plane can be partitioned into q parts of equal areas and perimeters provided q is a prime power.
The above questions we posed by A. Kaneko, M. Kano, R. Nandakumar, N. Ramana Rao, and I. Barany. These results are also inspired by the generalization of the BorsukUlam theorem by M. Gromov and Y. Memarian, in this paper I provide a theorem that further generalizes their result.
I also tried to deduce the theorem of Noga Alon about partitioning measures on the segment in a similar fashion, but did not succeed. By the way there arose a question about polynomials in one variable that may be interesting itself.

The discrete central point theorem claims the following: for a set X of n=(d+1)(r1)+1 points in ddimensional Euclidean space there exists a central point с such that any halfspace containing c contains at least r points of X.
A strong generalization of this theorem is the Tverberg theorem. The Tverberg theorem also has a generalization: the topological Tverberg theorem for continuous maps from (n1)dimensional simplex to ddimensional Euclidean space; the number r is required to be a prime power in this theorem.
In this paper a topological generalization of the central point theorem is proved for continuous maps from (n1)dimensional simplex to some ddimensional metric space (not necessarily Euclidean), here n=(d+1)(r1)+1 and r is arbitrary. Similar analogues of the Tvergerb theorem for maps to ddimensional metric spaces are considered; the number n has to be larger in such theorems.
If you do not have access to sciencedirect, you can download the paper from arxiv.org.

The Rattray theorem claims that every odd continuous map S^{n1}>S^{n1} maps some orthonormal basis to another orthonormal basis. Makeev used the underlying topological fact to prove another theorem: for any two absolutely continuous probabilistic measures in R^{n} there exists a set of n hyperplanes such that any two of them partition each measure into 4 equal parts.
In this paper we extend such results for the case of several maps or measures. Of course, in the Rattray type theorem we cannot obtain a basis, but we may ask for the larges possible k such that an orthonormal frame of k vectors is mapped to another orthonormal kframe by every map. The measure partition result should be changed accordingly; moreover, we prove a result that interpolates between the result of Makeev and the results of RamosZivaljevicVrecicaManiLevitska on measure partitions.
Another result that is worth mentioning is the “projective ham sandwich theorem” that relates partitioning of several measures by two hyperplanes in the projective space to the problem of embedding of this projective space to the Euclidean space.
This paper is also available at arxiv.org.

In this paper we prove the following fact, known as the GromovMilman conjecture: for every positive integer k and positive even d there exists a positive integer n=n(d,k) such that any polynomial of degree d in n variables equals (x_{1}^{2}+...+x_{n}^{2})^{d/2} after restriction to a kdimensional linear subspace of R^{n}.
We also consider similar results for odd degrees and complex polynomials (known as the Birch theorem), and improve the bounds for n(d,k) in such cases. Note that in the main case of even d we do not have the explicit value of n(d,k) at the moment.
The results proved by the ordinary cohomology calculations are due to Dol'nikov, the case of even degree is made by me, with use of the generalized BorsukUlam theorem for pgroups.
If you do not have access to SpringerLink, you can download the paper from arxiv.org.

In this paper I continue to study the analogues of the central point theorem for intersecting a family of convex sets by rays, see the previous paper on this subject.
In the previous paper the problem was: for a given family of convex compact sets, find a point such that every ray starting at this point intersects a large enough fraction of the family. Here we want a point such that any ray starting at this point misses a large enough fraction of the family. In the first case the sufficient condition was the dintersection property in R^{d}, in this case the sufficient condition is an upper bound for the covering multiplicity of the family.
Actually, in this paper the analogue of the central point theorem is deduced from the corresponding analogue of the Tverberg theorem, the latter is proved by topological methods.
This paper is also available at arxiv.org.

R.N. Karasev. “Z_{2}index of the grassmanian G_{2n}^{n}”, 2010.
The real Grassmann manifold G_{2n}^{n} of ndimensional linear subspaces in R^{2n} has a natural involution, given by the orthogonal complement. In this paper I estimate from above and from below the homological index of this involution. When n is a power of two, the estimates coincide and the index is 2n1.
These estimates have straightforward geometrical consequences. Let us give an example: if n is a power of two and K_{1},...,K_{2n1} are convex compact sets in R^{2n}, then there exist two orthogonal nsubspaces L and M, such that for any i the projections of K_{i} to L and M have equal measures.

R.N. Karasev. “Regular embeddings of manifolds and topology of configuration spaces”, 2010.
In this paper I again study the configuration spaces of manifolds, with applications to regular embeddings, that is continuous maps f : M>R^m such that images of every pairwise distinct k points are affinely (linearly) independent. In this work some new obstructions to existence of regular embeddings are given in terms of some characteristic classes of the tangent bundle of M.
I consider the subspace Q^q(M) of the qconfiguration space of a manifold M that can be identified with a bundle over M with fiber Q^q(R^m), where Q^q(R^m) is a certain submanifold of the qconfiguration space of R^m (m = dim M, q is a power of 2). The cohomology of Q^q(M) (or its parts) can be computed to give some explicit results on regular embeddings.
The explicit cohomology computations are also applied to the problem of multiple selfcoincidences of a continuous map between manifolds, which was studied in one of my previous papers.

In this paper we study affine planes of dimension m that intersect all members of a given family F of convex compacta. Such planes are called mtransversals.
The main lemma (Theorem 1) in this paper claims that if a family F has “many” mtransversals (in terms of nonvanishing of a certain cohomology class on the space of mtransversals), then F has at least one rtransversal. Here r
The above result (together with the Schubert calculus and the LusternikSchnirelmann category of the Grassmannian) allows to make some generalizations of the colorful Helly theorem to the case, when the number of colors if less than d+1 (d is the dimension of R^{d} in question).
My contribution to this paper is the proof of Theorem 1, some remarks on the colorfulHellytype theorems, and the case of complex vector space C^{d} and complex transversals.
This paper is also available at arxiv.org.

In this paper we prove two theorems. Informally, they claim that the nonzero elements of a finite field with odd characteristic can be partitioned into pairs with prescribed difference (maybe, with some alternatives) in each pair. We also consider some generalizations of these results to packing translates in a finite or infinite field.
My contribution to this paper is the topological proofs. Fedor Petrov has given the algebraic proofs for the partitions of F_{p}^{*} into pairs, and established several similar results with the same method.
If you have no access to Springerlink then you may download the paper from arxiv.org.

In this paper a measure of nonconvexity for a simple polygonal region in the plane is introduced. Informally it can be described as the minimal (negative) rotation of the tangent vector when traveling along the boundary anticlockwise. It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”.
If you do not have access to SpringerLink, you can download the paper from arxiv.org.

R.N. Karasev. “A note on Makeev's conjectures”, 2010.
The paper is accepted for publication in Journal of Mathematical Sciences.
In this paper the following modification of the Knaster problem is considered: describe all fourpoint sets X in a twodimensional sphere S such that for any continuous function f : S>R there exists a rotation that puts X into a level line of the function f.
V.V. Makeev conjectured that this claim is true for any four points, lying on a single circle, this is a necessary condition for linear f. Here a counterexample is given for this conjecture.
Another conjecture of Makeev claims that a quadrangle can be inscribed into any smooth closed simple curve on the plane iff it is inscribed into a circle.This conjecture does not have counterexamples by now, some new particular cases of this conjecture are solved positively.

R.N. Karasev. “Multiplicity of continuous maps between manifolds”, 2010.
In this paper the following question is studied: consider a continuous map f :M> N between two manifolds. Are there some sufficient conditions for existence of a coincident qtuple, i.e. a qtuple of pairwise distinct points x_1,..., x_q in M such that f(x_1) = f(x_2) = ... = f(x_q)?
I show that there are certain characteristic classes of the vector bundle f^*TNTM that guarantee the existence of a coincident qtuple for f, resembling similar results on singularities of smooth maps. In particular, I prove some nontrivial lower bound on the multiplicity for a continuous map of a real projective space of certain dimension into a Euclidean space of same or moderately larger dimension.
Some application to estimating the Krasnosel'skiiSchwarz genus of configuration spaces of manifolds are also given, continuing my previous results.

In this paper we study configurationlike spaces, i.e. the spaces of ordered qtuples of points in some topological space X with kwise conincidences forbidden. In fact we mostly work with the case of the Euclidean space X=R^n.
We study the measure of complexity for such spaces: the genus in the sense of Krasnosel'skiiSchwarzClappPuppe with respect to the action of some transitive permutation group on the qtuples. Here we mostly give the upper bounds, the lower bounds were studied in my paper in “Topology Apps”.
The upper bounds for the genus imply some results of BorsukUlamCohenLusk type for selfcoincidences of a continuous map, some versions of the Knaster problem are also considered.
This paper is also available from arxiv.org.

The paper was rejected in “Discrete and Computational Geometry”, the referee considered the main topological lemma known.
In this paper the colored KKM theorem is generalized to give an analogue of KKM theorem for the product of two simplices. Actually, an important particular case of this lemma (when the simplices have equal dimensions) was known before, the method of proof being quite wellknown. Still, the main lemma was not known before in the form, as stated in this paper.
This lemma has some corollaries, concerning cutting a family of figures in the plane by a set of straight lines, parallel to the coordinate axes.

This paper continues the series on inscribing crosspolytopes and partitioning measures into equal parts. The case of Z_2^k symmetry is considered here. It turns out that such a symmetry is not sufficient (unlike the case of (Z_p)^k for odd p). To obtain some results we need a larger noncommutative group of symmetry, isomorphic to D_8 \times (Z_2)^{k1}.
If you do not have access to SpringerLink, you can download the paper from arxiv.org.

R.N. Karasev. “Analogues of the central point theorem for families with dintersection property in R^{d}”, Combinatorica, 2013.
in this paper a family F of convex compact sets in R^{d} is considered, such that any d of the sets in F have a common point, this is the dintersection property, the minimal weakening of the condition of Helly's theorem. The analogues of the central point theorem and Tverberg's theorem for such families are proved:
A point p in R^{d} is found, such that any curve that passes through p and is unbounded, intersects at least about 1/(d+1) of the sets in F. Some analogues of the central transversal theorem are also proved.
The author's version is available at arxiv.org

Some results on the topology of the canonical bundle over the Grassmann variety are proved. They generalize the BorsukUlam theorem on coverings of a sphere and a ball. Some corollaries are deduced:
 Sufficient conditions for existence of a flat ktransversal for a family of n+1 compacts in R^n;
 Sufficient conditions for dividing n absolutely continuous probabilistic measures in R^n into parts of given size by a hyperplane;
 Some Hellytype theorems for flat transversals.
The English version of this paper is available at arxiv.org.

Some new cases of the Knaster conjecture on the functions on a sphere are proved. The set that is put into a level surface of the function is the orbit of a ptorus action with one point omitted. The proof is mainly based on the previous proofs of A.Yu. Volovikov for the Knaster conjecture for (full) ptori orbits.
This paper is also available at arxiv.org.

This paper develops the ideas of the paper about billiards (see above). The configurationlike spaces of smooth manifolds are considered. The lower bounds on the cohomological index i_G, introduced by A.Yu. Volovikov, on the genus in the sense of Krasnosel'skiiSchwarz, and the equivariant LyusternikSchnirelmann category of such spaces are given, with respect to the action of ptori by permuting points in the configuration.
The action of the symmetric group is also considered and some little generalization of the results of Fuks and Vassiliev on the Krasnosel'skiiSchwarz genus of the classical configuration space is proved.
The author's version can be downloaded from arxiv.org, if you do not have access to ScienceDirect.

In this paper the billiards in smooth convex bodies in R^n are considered. Consider the closed trajectories that make m reflections at the boundary per period. Using the LyusternikSchnirelmann theory, in the modification of Krasnosel'skiiSchwarz, I obtain the lower bound (n2)(m1)+2 for the number of distinct such trajectories, in the case of prime m.
This paper is also available at arxiv.org.

R.N. Karasev. “Inscribing a regular crosspolytope”. 2009.
The Russian version is also submitted to (and seems to be lost in) Mathematical Notes.
In this paper I prove that any smooth hypersurface S in R^{d}, which is diffeomorphic to an n1dimensional sphere, admits an inscribed crosspolytope (highdimensional octahedron) in the case, when n is an odd prime power. Possible weakenings of the smoothness condition are considered. The case of n=2^{k} is considered in a separate paper (see below).

In this paper a theorem on partitioning a measure in R^n into 2n equal parts by a system of cones is proved. The system of cones is obtained from some symmetrical system of cones, centered at the origin, by rigid motions. The results are mainly based on A.Yu. Volovikov's proof of the particular case of Knaster's problem on the functions on a sphere, where the orbits of ptori were considered.
The author's version can be downloaded from arxiv.org, if you do not have access to SpringerLink. In the published version the proof in Section 4 was incorrect; in this version it is corrected.

In this review some applications of algebraic topology to the combinatorial and convex geometry are considered. In particular, some author's results in the area are cited.
The author's version (in Russian) can be downloaded here.

In this paper the “dual” version of the central point theorem is considered:
Suppose n hyperplanes in general position are given in R^d. Then there exists a point x such that any ray starting at x intersects at least n/(d+1) of the given hyperplanes. Some generalizations of this result are also proved, that can be called the "dual" Tverberg theorem, and the "dual" central transversal theorem.
Besides, some generalization of the Brouwer fixed point theorem is proved for several fiberwise maps of a vector bundle into itself.
The published version of this paper contained an incorrect proof of the lemma on existence of a compact convex set that is mapped into itself by a given set of orthogonal projections onto affine hyperplanes. Later it was pointed out to me by Imre Barany that this lemma is known since 1984. The corrected English version is available at arxiv.org.

Helly's theorem states that a family of convex compact sets in R^d has nonempty intersection iff every its subfamily of size at most d+1 has nonempty intersection. In this paper we study the families such that every subfamily of size at most d has nonempty intersection, and try to find some upper bounds for the piercing number (see above).
A family of hyperplanes in general position shows that there cannot be a uniform upper bound for the piercing number, still the upper bounds can be found for families of balls, homothets of a single simplex, and some other families.
The author's version of this paper can be downloaded here, if you do not have access to SpringerLink.

In this paper we consider results of the following type:
Suppose a topological space X is mapped to a ddimensional metric space Y. Then there exists a point p in Y such that the LyusternikSchnirelmann category of its preimage (or a neighborhood of its preimage) in X is at least the category of X, divided by d+1. It is proved that such results hold for other integer functions on open subsets of topological spaces, that are similar to the LyusternikSchnirelmann category in the sense of three axioms (monotonicity, subadditivity, and the upper bound for the disjoint union).
The corrected English version of this paper is available at arxiv.org.

In this paper the Tverberg transversal conjecture and its generalizations are considered. Some particular cases of this conjecture are proved. The Tverberg transversal conjecture is stated as follows:
Suppose that m+1 finite sets S_0, ... , S_m in R^d are given. Then each set S_i can be partitioned into r_i parts S_i1, ... , S_ir_i so that the convex hulls of all the sets S_ij can be intersected by a single mflat (affine subspace of dimension m), such a flat is called mtransversal. The requirement that the cardinality of S_i is at least (r_i1)(dm+1) + 1 is also needed.
In this paper this conjecture is proved for the case, when the numbers r_i are powers of the same prime. If this prime is odd, than the number dm is required to be even. Other cases of this conjecture, for example when the numbers r_i are equal to the same prime, were known before.
The author's version can be downloaded here, if you do not have access to SpringerLink.

In this paper the results on the conjecture of A. Bezdek, and polytope partitiones are generalized. Actually, a "colored" generalization of the KnasterKuratowskiMazurkiewicz theorem (KKM) theorem is proved, which implies some of the previous results directly.

In this paper the ideas around the problem of Proizvolov and the cojecture of A. Bezdek are developed. In particular, it is proved that any convex polytope P of unit volume with facets F_i can be partitioned into convex sets A_i so that the following holds:
 The intersection of each A_i with the boundary of P equals F_i;
 The volumes of A_i are equal to the given positive numbers with unit sum.
The author's version can be downloaded here, if you do not have access to SpringerLink.

This paper is a piece of PhD thesis, where two definitions of Mstrong convexity are compared. It is shown that they are equivalent for generating sets M.

The thesis is mainly composed of the previous publications. The chapter about the strong convexity, generating sets, and bodies of constant width in Banach spaces is rewritten, some of the results are formaluted for any Banach space instead of R^n.

In this paper I prove some generalizations of the following wellknown problem (as far as I known, by Proizvolov). Let P be a convex ngon in the plane, and suppose n points inside P are given. Then there exists a bijection between the points and the edges of P, such that all the corresponding triangles (formed by a point and its respective edge) cover P. There is another version, when triangles are required to be pairwise disjoint, instead of covering P. The problem has generalizations to higher dimensions.
The question of A. Bezdek was: what is changed if the n points are not required to be inside P. This paper answers this question, along with some other results.

In this paper the notion of Mstrongly convex set is studied. Let M be a closed convex set, then any intersection of a family of translates of M is called Mstrongly convex set. It was shown before that Mstrongly convex sets behave well, if the set M is generating (see above). In particular, in this paper an analogue of the Caratheodory theorem is established for Mstrong covexity.
It should be mentioned, that in this paper a lemma was proved, that could be of its own interest. Let A and B be convex compact sets in R^n, then the set difference of A and A+B (here is the Minkowski sum) is a contractible set.

In this paper the following type of closed convex sets is studied. The closed convex set M is called "generating" if any intersection of a finite number of translates of M is a summand of M w.r.t. the Minkowski sum. There is a large number of examples of generating sets, unit balls are generating, and any Cartesian product of generating sets is again generating.
The main result of this paper claims that in the definition of generating sets it suffices to consider intersections of two translates instead of arbitrary number of translates.

In this paper a conjecture of Grunbaum is proved. Consider a family F of translates of a convex compact set in the plane, such that every two members of the family have a common point. Then the family F can be pierced by three points.
The term "pierced" means that a set T of three points exists, such that any set K of the family F has nonempty intersection with T. In general, a point set, that "pierces" a family of sets F is called a "transversal" of F, the minimal size of a transversal for F is called the "piercing number" of F.
The results were also published in Russian in Proceedings of the seminars at St.Petersburg Department of the Mathematical Institute.