On the hollow enclosed by convex sets

Abstract: For

n l e q d

, a family

c a l F = C_{0}, C_{1}, l d o t s, C_{n}

of compact convex sets in

R^{d}

is called an

n

-critical family provided any

n

members of

c a l F

have a non-empty intersection, but

. 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

Here we prove that the closure of the convex hull of a hollow in

R^{d}

is a

d

-simplex.