Abstract
For brevity, we call a family $ = [K Id e I] of closed convex sets, in d-dimensional euclidean space E d, an intersectional configuration of class n if $ satisfies the following conditions: (i) The intersection ]( = N [K~I~ e I] is a convex body set with a nonempty interior) and K is a proper subset of each K . (ii) There exists an integer m ~ 2 such that the intersection of any m members of ~ is K. The class n is the smallest such value of m. L.M. Blumenthal (private communication) pointed out to the author that, for d = n = 2, an intersectional configuration with three members has the property that if a convex set B intersects all three members of ~ then B must intersect K. In this paper, we extend this result to the general case by proving the following theorem. Theorem. If a convex set B intersects dn d + 1 or more members of an intersectional configuration $, then B intersects
Published Version
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