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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2008

1 Stavrakas N.

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Measure and Helly's Intersection Theorem for Convex Sets

100%

Stavrakas N.

nr 1

59-65

Let $ℱ = {F_α}$ be a uniformly bounded collection of compact convex sets in ℝ ⁿ. Katchalski extended Helly's theorem by proving for finite ℱ that dim (⋂ ℱ) ≥ d, 0 ≤ d ≤ n, if and only if the intersection of any f(n,d) elements has dimension at least d where f(n,0) = n+1 = f(n,n) and f(n,d) = max{n+1,2n-2d+2} for 1 ≤ d ≤ n-1. An equivalent statement of Katchalski's result for finite ℱ is that there exists δ > 0 such that the intersection of any f(n,d) elements of ℱ contains a d-dimensional ball of measure δ where f(n,0) = n+1 = f(n,n) and f(n,d) = max{n+1,2n-2d+2} for 1 ≤ d ≤ n-1. It is proven that this result holds if the word finite is omitted and extends a result of Breen in which f(n,0) = n+1 = f(n,n) and f(n,d) = 2n for 1 ≤ d ≤ n-1. This is applied to give necessary and sufficient conditions for the concepts of "visibility" and "clear visibility" to coincide for continua in ℝ ⁿ without any local connectivity conditions.