A measure of convexity for compact sets

Rolf Schneider

Vol. 58 (1975), No. 2, 617–625

DOI: 10.2140/pjm.1975.58.617

Abstract

For a subset M of d-dimensional real vector space R^d let c(M) = inf{λ ≧ 0|M + λ conv M is convex}, where conv M is the convex hull of M and + denotes vector addition of sets. Among the compact subsets of R^d, the convex sets are characterized by the equality c(M) = 0. It is proved that c(M) ≦ d for arbitrary subsets of R^d, with equality if and only if M consists of d + 1 affinely independent points. If M is either unbounded or connected, then c(M) ≦ d − 1; the bound d − 1 is best possible in either case.

Mathematical Subject Classification 2000

Primary: 52A20

Milestones

Received: 25 June 1974

Published: 1 June 1975