|
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. ==Dual cone== The dual cone ''C *'' of a subset ''C'' in a linear space ''X'', e.g. Euclidean space R''n'', with topological dual space ''X *'' is the set : where <''y'', ''x''> is the duality pairing between ''X'' and ''X *'', i.e. <''y'', ''x''> = ''y''(''x''). ''C *'' is always a convex cone, even if ''C'' is neither convex nor a cone. Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as R''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''. : Using this latter definition for ''C *'', we have that when ''C'' is a cone, the following properties hold: * A non-zero vector ''y'' is in ''C *'' if and only if both of the following conditions hold: #''y'' is a normal at the origin of a hyperplane that supports ''C''. #''y'' and ''C'' lie on the same side of that supporting hyperplane. *''C *'' is closed and convex. *''C''1 ⊆ ''C''2 implies . *If ''C'' has nonempty interior, then ''C *'' is ''pointed'', i.e. ''C *'' contains no line in its entirety. *If ''C'' is a cone and the closure of ''C'' is pointed, then ''C *'' has nonempty interior. *''C * *'' is the closure of the smallest convex cone containing ''C'' (a consequence of the hyperplane separation theorem) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dual cone and polar cone」の詳細全文を読む スポンサード リンク
|