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 ^n+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 <sup>\prime</sup> 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 .

Authors and Affiliations

1. Instituto de Matemáticas, UNAM, Circuito exterior, Ciudad Universitaria, C.P. 04510, México DF, Mexico, Mexico

   J. Bracho & L. Montejano

Received October 25, 2000, and in revised form September 27, 2001, and October 17, 2001. Online publication March 1, 2002.

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