It is shown that the convex
kernel of a compact star-shaped subset S of a finite-dimensional linear topological
space Ln is determined by the (n− 1)-extreme points of S. The cardinality of the set
of k-extreme points is determined for compact star-shaped sets of dimension greater
than two. Also given is the result that any compact star-shaped subset S of Ln
contains a countable set of (n − 1)-extreme points which determines the convex
kernel of S. Another result is that a compact nonconvex star-shaped set S
in a locally convex space L is determined by the convex kernel of S and
the subset of points that are extreme in S relative to the convex kernel of
S.
