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 = 〈K₁,..., K_n〉. A family ℱ of (n, d)-bodies is weakly intersecting if there exists an n-point p = 〈p₁,..., p_n〉 such that for every K ∈ ℱ there exists an index 1 ≤ i ≤ n for which p_i ∈ K_i. A family ℱ of (n, d)-bodies is strongly intersecting if there exists an index i such that ∩_K∈ℱ K_i ≠ ∅. 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) ≤ ([log₂(n + 1)] + 1)^d. In addition, we introduce and discuss two interesting related questions of a combinatorial-topological nature.