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.
随便看	core of a subgroup CornerOfARing corner of a ring CornerPointTheorem corner point theorem CorollariesOfBasicTheoremOnOrderedGroups corollaries of basic theorem on ordered groups CorollaryOfBanachAlaogluTheorem corollary of Banach-Alaoglu theorem CorollaryOfBezoutsLemma CorollaryOfBorelCantelliLemma corollary of Borel-Cantelli lemma corollary of Bézout’s lemma CorollaryOfCauchyIntegralTheorem corollary of Cauchy integral theorem CorollaryOfEulerFermatTheorem corollary of Euler-Fermat theorem CorollaryOfKummersTheorem corollary of Kummer’s theorem CorollaryOfMorleysTheorem corollary of Morley’s theorem CorollaryOfSchurDecomposition corollary of Schur decomposition CorollaryToTheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois corollary to the compositum of a Galois extension and another extension is Galois