On starshaped sets and Helly-type theorems

John David Baildon and Ruth Silverman

Vol. 62 (1976), No. 1, 37–41

DOI: 10.2140/pjm.1976.62.37

Abstract

Suppose an ordered pair of sets (S,K) in a linear topological space is of Helly type (n + 1,n), i.e., for every n + 1 distinct points in S there is a point in K which sees at least n of them via S. Then if S is closed, K compact, and n ≧ 3, the nontrivial visibility sets in K are pairwise nondisjoint. Sufficient conditions are obtained for S to be starshaped.

Mathematical Subject Classification 2000

Primary: 52A30

Secondary: 52A35

Milestones

Received: 2 July 1975

Revised: 9 October 1975

Published: 1 January 1976