Note on Star-shaped Sets

1. The aim of this note is to prove that if 717" is a compact subset of En and for some m every w-dimensional hyperplane through a fixed point pQEn intersects M along a nonempty acyclic set, lúm^n — 1, then M is star-shaped with respect to p, i.e., if aQM then the segment pa is contained in M. This theorem is a generalization of a theorem of Aumann [l]. We gave recently a proof of Aumann's theorem based on the theory of multivalent mappings (see [3]); the present proof follows essentially the same line. The topological lemma on which it is based is susceptible to further generalizations; however, we give it here only in its simplest and easily proved case which is needed for the proof of the theorem about star-shaped sets. I have the pleasure to acknowledge my indebtedness to Dr. M. W. Hirsch for valuable suggestions.