On the hollow enclosed by convex sets

Abstract

For $n\le d$, a family $\mathcal{F}=\{C_0,C_1,\dots ,C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of $\mathcal{F}$ have a non-empty intersection, but ${\bigcap }_{i=0}^n{C}_i=\varnothing$. If $n=d$ then a lemma on the intersection of convex sets due to Klee implies that the $d+1$ members of the $d$-critical family enclose a `hollow' in $R^d$, a bounded connected component of $R^d\setminus {\bigcup }_{i=0}^n{C}_i.$ Here we prove that the closure of the convex hull of a hollow in $R^d$ is a $d$-simplex.

7 pages. Identical to original version, just some new acknowledgements