# exposed points are dense in the extreme points

###### Definition.

Let $K\subset{\mathbb{R}}^{n}$ 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 $K\subset{\mathbb{R}}^{n}$ 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 ${\mathbb{R}}^{2}$ 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 ExposedPointsAreDenseInTheExtremePoints 2013-03-22 17:41:04 2013-03-22 17:41:04 jirka (4157) jirka (4157) 4 jirka (4157) Theorem msc 52A99 Strasziewicz theorem ExtremePoint exposed point