Tverberg-Type Theorems for Intersecting by Rays

Abstract

Abstract. In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set.

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

“…Though we give a separate proof for Theorem 1.5 to clarify the exposition. The reasoning in this proof (and the subsequent proofs) is essentially the same as in [11,12].…”

“…While this paper was considered and reviewed in the journal, another paper [12] with similar results was published. So the content of this paper has a large intersection with that of [12]. The author thanks V.L.…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…It turns out that many of the discussed depth measures are special cases of the following more general conjecture, first proposed by Mustafa et al [296]. See also the related paper [225]. The case k = 0 corresponds to halfspace depth, k = d to simplicial depth, and k = d − 1 to ray-shooting depth.…”

“…It turns out that many of the discussed depth measures are special cases of the following more general conjecture, first proposed by Mustafa et al [296]. See also the related paper [225]. It is not hard to show that given a set P of n points in R d , and an integer 0 ≤ k ≤ d − 1, there exists a point q ∈ R d such that any (d − k)-half flat through q intersects at least max…”

A topological central point theorem

“…Without loss of generality, let g −1 (F ) be defined by the equations Recall the known fact: The case W = R d of Theorem 1.1 follows from the topological Tverberg theorem (only the case of prime r is needed). For the reader's convenience we present a proof here (see also [7,Section 6]).…”