A measure of convexity for compact sets

For a subset M of d-dimensional real vector space R d let c (M) = inf { ^ 0|M + conv M is convex), where convM 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 jR 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.