exposed points are dense in the extreme points

Definition.

Let $K\subset {ℝ}^n$ be a closed convex set. A point $p\in K$ is called an exposed point if there is an $n-1$ dimensional hyperplane whose intersection with $K$ is $p$ alone.

Theorem (Strasziewicz).

Let $K\mathrm{\subset }{\mathrm{R}}^n$ be a closed convex set. Then the set of exposed points is dense in the set of extreme points.

For example, let $C(p)$ denote the closed ball in ${ℝ}^2$ of radius 1 and centered at $p.$ Then take $K$ to be the convex hull 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