Measure and Helly's Intersection Theorem for Convex Sets
Volume 56 / 2008
Abstract
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.
