Exposed point について

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Exposed point ：ウィキペディア英語版

Exposed point
In mathematics, an exposed point of a convex set

C

is a point

x\in C

at which some continuous linear functional attains its strict maximum over

C

. Such a functional is then said to ''expose''

x

. Note that there can be many exposing functionals for

x

. The set of exposed points of

C

is usually denoted

\exp(C)

.
A stronger notion is that of ''strongly exposed point'' of

C

which is an exposed point

x \in C

such that some exposing functional

f

x

attains its strong maximum over

C

x

, i.e. for each sequence

(x_n) \subset C

we have the following implication:

f(x_n) \to \max f(C) \Longrightarrow \|x_n -x\| \to 0

. The set of all strongly exposed points of

C

is usually denoted

\operatorname\exp(C)

.
There are two weaker notions, that of extreme point and that of support point of

C

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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