Helly-Type Theorems on the Homology of the Space of Transversals

In this paper we ``measure' the size of the set of n -transversals of a family F of convex sets in R <sup>n+k</sup> according to its homological complexity inside the corresponding Grassmannian manifold. Our main result states that the ``measure' μ of the set of n -transversals of F is greater than or equal to k if and only if every k+1 members of F have a common point and also if and only if for some integer m , 1≤ m≤ n , and every subfamily F ^prime of F with k+2 members, the ``measure' μ of the set of m -transversals of F ^prime is greater than or equal to k . Received October 25, 2000, and in revised form September 27, 2001, and October 17, 2001. Online publication March 1, 2002.