Let 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) = maxn+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) = maxn+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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-7,
author = {N. Stavrakas},
title = {Measure and Helly's Intersection Theorem for Convex Sets},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {56},
year = {2008},
pages = {59-65},
zbl = {1142.52009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-7}
}
N. Stavrakas. Measure and Helly's Intersection Theorem for Convex Sets. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) pp. 59-65. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-1-7/