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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert metric ： ウィキペディア英語版 Hilbert metric In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the ''n''-dimensional Euclidean space R''n''. It was introduced by as a generalization of the Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the ''n''-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces. == Definition == Let Ω be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points ''A'' and ''B'' of Ω, let ''X'' and ''Y'' be the points at which the straight line ''AB'' intersects the boundary of Ω, where the order of the points is ''X'', ''A'', ''B'', ''Y''. Then the Hilbert distance ''d''(''A'', ''B'') is the logarithm of the cross-ratio of this quadruple of points: : $d(A,B)=\backslash log\backslash left(\backslash frac\backslash frac\backslash right).$ The function ''d'' is extended to all pairs of points by letting ''d''(''A'', ''A'') = 0 and defines a metric on Ω. If one of the points ''A'' and ''B'' lies on the boundary of Ω then ''d'' can be formally defined to be +∞, corresponding to a limiting case of the above formula when one of the denominators is zero. A variant of this construction arises for a closed convex cone ''K'' in a Banach space ''V'' (possibly, infinite-dimensional). In addition, the cone ''K'' is assumed to be ''pointed'', i.e. ''K'' ∩ (−''K'') =  and thus ''K'' determines a partial order $\backslash leq\_K$ on ''V''. Given any vectors ''v'' and ''w'' in ''K'' \ , one first defines : $M(v/w)=\backslash inf\backslash ,\; \backslash quad\; m(v/w)=\backslash sup\backslash .$ The Hilbert pseudometric on ''K'' \  is then defined by the formula : $d(v,w)=\backslash log\backslash frac.$ It is invariant under the rescaling of ''v'' and ''w'' by positive constants and so descends to a metric on the space of rays of ''K'', which is interpreted as the projectivization of ''K'' (in order for ''d'' to be finite, one needs to restrict to the interior of ''K''). Moreover, if ''K'' ⊂ R × ''V'' is the cone over a convex set Ω, : $K=\backslash ,$ then the space of rays of ''K'' is canonically isomorphic to Ω. If ''v'' and ''w'' are vectors in rays in ''K'' corresponding to the points ''A'', ''B'' ∈ Ω then these two formulas for ''d'' yield the same value of the distance. 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「Hilbert metric」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.