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In this paper we \"measure\" the size of the set of n -transversals of a family F of convex sets in Rn+k 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 .

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