Supporting hyperplane について

has at least one boundary-point on the hyperplane.
Here, a closed half-space is the half-space that includes the points within the hyperplane.
==Supporting hyperplane theorem==

This theorem states that if

S

is a convex set in the topological vector space

X=\mathbb^n,

and

x_0

is a point on the boundary of

S,

then there exists a supporting hyperplane containing

x_0.

x^* \in X^* \backslash \

(

X^*

is the dual space of

X

x^*

is a nonzero linear functional) such that

x^*\left(x_0\right) \geq x^*(x)

for all

x \in S

, then
:

H = \

defines a supporting hyperplane.
Conversely, if

S

is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then

S

is a convex set.〔
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set

S

is not convex, the statement of the theorem is not true at all points on the boundary of

S,

as illustrated in the third picture on the right.
The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.〔Cassels, John W. S. (1997), ''An Introduction to the Geometry of Numbers'', Springer Classics in Mathematics (reprint of 1959() and 1971 Springer-Verlag ed.), Springer-Verlag.〕
A related result is the separating hyperplane theorem, that every two disjoint convex sets can be separated by a hyperplane.

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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