Measure and Helly's Intersection Theorem for Convex Sets

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 intersec