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.