Theorems of Borsuk-Ulam type for flats and common transversals of families of convex compact sets

Abstract

Abstract. In this paper some results on the topology of the space of k-flats in R n are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in R n , and on measure partitions by hyperplanes, are deduced.

Topological transversals to a family of convex sets

“…has already given useful geometric applications to transversal planes in [16,8,17]. Several results on transversals, similar to the results of this paper, can be found in [2,4,5,11]. In [13] Theorem 1 was conjectured and verified in some low-dimensional cases.…”

“…It is natural to ask, what happens if the number of colors is less than d + 1. Some results of this kind were already established in [11,Theorems 21,22,23]. We are going to prove more results in this direction.…”

Tverberg-Type Theorems for Intersecting by Rays

“…We need the following lemma, that can be considered a strengthening of the definition of genus. A particular case of this lemma for G = Z 2 was proved in [12,Theorem 9].…”

Cutting the Same Fraction of Several Measures

“…Note that in [2,6,8] a similar problem was considered: Cut by a hyperplane a prescribed fraction of each of d measures in R d . Again, this cannot be done in general and some additional assumptions were needed.…”