Analogues of the central point theorem for families with d-intersection property in ℝ d

Abstract

In this paper we consider families of compact convex sets in R d such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg's theorem for such families.2000 Mathematics Subject Classification. 52A20, 52A35, 52C35.

Tverberg-Type Theorems for Intersecting by Rays

“…, and e(f ) = 0. Similar to the proof of Lemma 3 in [11], we conclude that the natural map A l G → H l G (Z) is injective in dimensions l ≤ n − r − (r − 1)d = n − 1 − (r − 1)(d + 1). Let us sketch the proof of this claim.…”

“…This paper reproduces some lemmas from [11]. In Tverberg's theorem and its topological generalizations (see [2,24] for example) it is important to consider the configuration space of r-tuples of points x 1 , .…”

“…In this section some topological facts, that arise in the proof of Theorem 1 are given. In fact, the first part of the proof follows the proof of Theorem 1.1 in [11], this and the following sections restate the needed lemmas.…”

“…Denote the zero set of f by Z. Similar to [11], the map f can be considered as a section of G-equivariant vector bundle, its Euler class being…”

“…The point p divides ℓ into two half-lines, we call these half-lines rays starting at p. We are going to study the questions of the following type: given a family F of convex sets in R d , find a point p ∈ R d such that every ray starting at p intersects at least α|F | members of F , or at most β|F | members of F . Such questions were considered before in [20,10], for the case of hyperplanes, and in [5,11] for families of convex sets.…”