Transversal, Helly and Tverberg type Theorems in Geometry, Combinatorics and Topology III

Helly’s Theorem and its relatives, the theorems of Radon and Caratheodory have played an important role in Discrete and Convex Geometry, there are numerous generalizations and variations of them that have been studied from different perspectives and points of view. The classical lemma of Radon for instance states that any d + 2 points in R can be partitioned into two classes with intersecting convex hulls. This theorem can be rephrased in the following equivalent way: Given any affine map: f : ∆ → R, been ∆ the d + 1 simplex in R there are two disjoint faces σ1 and σ2 of ∆ such that f(σ1) ∩ f(σ2) 6= ∅. During this workshop one of the main philosophical aspects was related with the question of up to what extent this classical theorems really depend on convexity or linearity, i.e., whether or not there are some more fundamental topological principles that underly them For instance topological generalizations of Helly’s theorem, topological generalizations of the classical theorems in discrete and convex geometry of BorosFüredi (d=2) and of Bárány. One of the most beautiful theorems in combinatorial convexity is due to Tverberg, that is the r-partite version of Radon’s Lemma. To be more precise,Tverberg’s theorem states that every (d+1)(r−1)+1 points in Euclidean d-space R can be partitioned into r parts such that the convex hulls of these parts have nonempty intersection. This theorem still remains central and is one of the most intriguing results of combinatorial geometry. It has been shown that there are many close relations between Tverberg’s theorem and several important results in mathematics, such as; Rado’s Central Theorem on general measures, the Ham Sandwich Theorem and the Four Color Theorem, just to mention some examples. The original Tverberg Theorem now has several different proofs, including those by Tverberg, Roudneff, Sarkaria and more recently due to Zvagelśkii. A specially elegant proof is due to Zarkaria with further simplifications of Onn. And could be rephrased in the following way.