Realisation of separoids and a Tverberg-type problem

A separoid is a symmetric relation † ⊂ ( 2 2 ) defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B C † D ⇐⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a family of (convex) polytopes, and their Radon partitions, in the Euclidean space of dimension |S| − 1. Furthermore, we introduce a new kind of separoids’ morphisms — called chromomorphisms— which allow us to study Tverberg’s generalisation (1966) of Radon’s theorem (1921) in the context of convex sets. In particular the following Tverberg-type theorem is proved: Let S be a separoid of order |S| = (k − 1)(d(S) + 1) + 1, where d(S) denotes the (combinatorial) dimension of S. If there exists a monomorphism S → P into a separoid of points in general position in IE, then there exists a chromomorphism S −→ Kk onto the complete separoid of order k. This theorem is, in a sense, dual to the Hadwiger-type theorem proved by Arocha, Bracho, Montejano, Oliveros & Strausz [Disc. & Comp. Geom., 27, 2002, 377-385].