Measure and Helly's Intersection Theorem for Convex Sets

N. Stavrakas¹

¹ Department of Mathematics University of North Carolina Charlotte, NC 28223, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 1, pp. 59-65

Résumé

Let ${\cal F}=\{F_\alpha \}$ be a uniformly bounded collection of compact convex sets in $\mathbb R^n$. Katchalski extended Helly's theorem by proving for finite ${\cal F}$ that $\dim (\bigcap {\cal F})\geq d$, $0\leq d\leq n$, 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 $1\leq d\leq n-1.$ An equivalent statement of Katchalski's result for finite ${\cal % F}$ is that there exists $\delta >0$ such that the intersection of any $% f(n,d)$ elements of ${\cal F}$ contains a $d$-dimensional ball of measure $% \delta $ where $f(n,0)=n+1=f(n,n)$ and $f(n,d)=\max \{n+1,2n-2d+2\}$ for $% 1\leq d\leq n-1.$ 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 $1\leq d\leq n-1$. This is applied to give necessary and sufficient conditions for the concepts of “visibility” and “clear visibility” to coincide for continua in $\mathbb R^n$ without any local connectivity conditions.

Détail

DOI : 10.4064/ba56-1-7

Keywords: cal alpha uniformly bounded collection compact convex sets mathbb katchalski extended hellys theorem proving finite cal dim bigcap cal geq leq leq only intersection elements has dimension least where max n leq leq n equivalent statement katchalskis result finite cal there exists delta intersection elements cal contains d dimensional ball measure delta where max n leq leq n proven result holds word finite omitted extends result breen which leq leq n applied necessary sufficient conditions concepts visibility clear visibility coincide continua mathbb without local connectivity conditions