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 exposedpoint 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 setof 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 $(1,-1)$ areextreme points, but they are not exposed points.

单词	ExposedPointsAreDenseInTheExtremePoints
释义	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 exposedpoint 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 setof 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 $(1,-1)$ areextreme points, but they are not exposed points.
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