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

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 of (n,d)-bodiesis weakly intersecting if there exists an n -point p = < p ₁ , . . . , p _n > such that for every there exists an index 1 for which p _i ∈ K _i . A family of (n,d)-bodies is strongly intersecting if there exists an index i such that . 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 . In addition, we introduce and discuss two interesting related questions of a combinatorial-topological nature. Received February 9, 1996, and in revised form November 6, 1996, December 16, 1996, and January 7, 1998.