Vol. 58, No. 2, 1975

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A measure of convexity for compact sets

Rolf Schneider

Vol. 58 (1975), No. 2, 617–625
Abstract

For a subset M of d-dimensional real vector space Rd 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 Rd, the convex sets are characterized by the equality c(M) = 0. It is proved that c(M) d for arbitrary subsets of Rd, 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
Authors
Rolf Schneider
Mathematisches Institut
Albert-Ludwigs-Universität Freiburg
Eckerstr. 1
D-79104 Freiburg im Breisgau
Germany