Helly-Type Theorems on the Homology of the Space of Transversals |
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Authors: | J. Bracho L. Montejano |
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Affiliation: | (1) Instituto de Matemáticas, UNAM, Circuito exterior, Ciudad Universitaria, C.P. 04510, México DF, Mexico, MX |
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Abstract: | 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 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. |
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