Tverberg’s theorem is 50 years old: A survey

Abstract

This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. r 1 (z − y j ) = 0. Note that z = y j is possible but cannot hold for all j since µ > 0.Define Y j ⊂ X j for j = 1, . . . , r via y j ∈ relint conv Y j . We claim that r 1 aff Y j = ∅. Otherwise there is a point v ∈ r 1 aff Y j . Let ·, · denote the standard scalar product, so x, x = x 2 , for instance. Then z − v, z − y j > 0 if y i = z (because y j is the closest point to z in conv Y j ) Imre Bárány

Inscribed Tverberg-Type Partitions for Orbit Polytopes

“…The result is of fundamental importance in discrete and convex geometry, with a panoply of rich generalizations and variants in geometric combinatorics and beyond. We refer the reader to the surveys [3,5,10] for the history of Tverberg's theorem and its relatives, as well as a sampling of its many applications.…”

Radon Numbers Grow Linearly

“…• k to r k ≤ c • r • k. This would truly lead to an improvement of the upper bound on f in Theorem 1.2 and would lead to further applications [6]. by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.…”

“…1 It was also shown in [16] that it follows from the work of Alon et al [2] that weak ε-nets [1] of size c(ε, r ) also exist and a ( p, q)-theorem [3] also holds, so understanding these parameters better might lead to improved ε-net bounds. It remains an interesting challenge and a popular topic to find new connections among such theorems; for some recent papers studying the Radon numbers or Tverberg theorems of various convexity spaces, see [9][10][11]14,20,23,24,27], while for a comprehensive survey, see Bárány and Soberón [6].…”

An application of the universality theorem for Tverberg partitions to data depth and hitting convex sets

“…In this paper we present an application of the following statement, which is one consequence of a recent theorem of Pór (see [1]). We use Lemma 1 to prove the following theorem.…”