Intersection Properties of Families of Convex (n,d)-Bodies

Abstract

We study a multicomponent generalization of Helly's theorem. An (n,d)-body K is an ordered n -tuple of d -dimensional sets, K= < K 1 , . . . ,K n > . A family $\cal F$ of (n,d)-bodiesis weakly intersecting if there exists an n -point p = < p 1 , . . . , p n > such that for every $K \in {\cal F}$ there exists an index 1 $\leq i \leq n$ for which p i ∈ K i . A family $\cal F$ of (n,d)-bodies is strongly intersecting if there exists an index i such that $\bigcap_{K\in{\cal F}} K_i \neq \emptyset$ . The main question addressed in this paper is: What is the smallest number H(n,d), such that for every finite family of convex (n,d)-bodies, if every H(n,d) of them are strongly intersecting, then the entire family is weakly intersecting? We establish some basic facts about H(n,d) , and also prove an upper bound $H(n,d)\leq (\lfloor \log_2 (n+1) \rfloor + 1)^d$ . In addition, we introduce and discuss two interesting related questions of a combinatorial-topological nature.