A short and almost elementary proof of the Borosâ€“FÃ¼rediâ€“BÃ¡rÃ¡nyâ€“Pachâ€“ Gromov theorem on the multiplicity of covering by simplices in R d is given. Let us give a proof of theâ€¦ (More)

In this paper, we consider finite families of convex sets in R such that every d or fewer sets of the family have a common point. For some families of this type, we give upper bounds on the size of aâ€¦ (More)

In this paper we prove that it is possible to inscribe a regular crosspolytope (multidimensional octahedron) into a smooth convex body in R, where d is an odd prime power. Some generalizations ofâ€¦ (More)

Introduction One possible way to express that the cube Q d = [0, 1] d has dimension d is to notice that it cannot be colored in d colors with arbitrarily small connected monochromatic components. Aâ€¦ (More)

For convex partitions of a convex body B we prove that we can put a homothetic copy of B into each set of the partition so that the sum of homothety coefficients is â‰¥ 1. In the plane the partitionâ€¦ (More)

We prove some analogues of the central point theorem and Tverbergâ€™s theorem, where instead of points, we consider hyperplanes or affine flats of given dimension.

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahlerâ€™s conjecture on theâ€¦ (More)