Vol. 29, No. 2, 1969

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Extremal structure of star-shaped sets

Freddie Eugene Tidmore

Vol. 29 (1969), No. 2, 461–465

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 (n1)-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.

Mathematical Subject Classification
Primary: 46.01
Secondary: 52.00
Received: 13 May 1968
Published: 1 May 1969
Freddie Eugene Tidmore