Measure and Helly's Intersection Theorem for Convex Sets

N. Stavrakas. "Measure and Helly's Intersection Theorem for Convex Sets." Bulletin of the Polish Academy of Sciences. Mathematics 56.1 (2008): 59-65. <http://eudml.org/doc/281319>.

@article{N2008,
abstract = {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) = 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.},
author = {N. Stavrakas},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Helly's theorem; cone; visibility},
language = {eng},
number = {1},
pages = {59-65},
title = {Measure and Helly's Intersection Theorem for Convex Sets},
url = {http://eudml.org/doc/281319},
volume = {56},
year = {2008},
}

TY - JOUR
AU - N. Stavrakas
TI - Measure and Helly's Intersection Theorem for Convex Sets
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 1
SP - 59
EP - 65
AB - 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) = 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.
LA - eng
KW - Helly's theorem; cone; visibility
UR - http://eudml.org/doc/281319
ER -