exposed points are dense in the extreme points


Let Kn be a closed convex set. A point pK is called an exposed point if there is an n-1 dimensional hyperplaneMathworldPlanetmath whose intersection with K is p alone.

Theorem (Strasziewicz).

Let KRn be a closed convex set. Then the set of exposed points is dense in the set of extreme pointsPlanetmathPlanetmath.

For example, let C(p) denote the closed ball in 2 of radius 1 and centered at p. Then take K to be the convex hullMathworldPlanetmath of C(-1,0) and C(1,0). The points (-1,1), (-1,-1), (1,1), and (1,-1) are extreme points, but they are not exposed points.

Title exposed points are dense in the extreme points
Canonical name ExposedPointsAreDenseInTheExtremePoints
Date of creation 2013-03-22 17:41:04
Last modified on 2013-03-22 17:41:04
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Theorem
Classification msc 52A99
Synonym Strasziewicz theorem
Related topic ExtremePoint
Defines exposed point