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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 intrinsic metric ： ウィキペディア英語版 intrinsic metric In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from one point to another along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the length of all paths from one point to the other. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space. ==Definitions== Let $(M,\; d)$ be a metric space. We define a new metric $d\_\backslash text$ on $M$, known as the induced intrinsic metric, as follows: $d\_\backslash text(x,y)$ is the infimum of the lengths of all paths from $x$ to $y$. Here, a ''path'' from $x$ to $y$ is a continuous map :$\backslash gamma\; \backslash colon\; ()\; \backslash rightarrow\; M$ with $\backslash gamma(0)\; =\; x$ and $\backslash gamma(1)\; =\; y$. The ''length'' of such a path is defined as explained for rectifiable curves. We set $d\_\backslash text(x,y)\; =\backslash infty$ if there is no path of finite length from $x$ to $y$. If :$d\_\backslash text(x,y)=d(x,y)$ for all points $x$ and $y$ in $M$, we say that $(M,\; d)$ is a length space or a path metric space and the metric $d$ is intrinsic. We say that the metric $d$ has approximate midpoints if for any $\backslash varepsilon>0$ and any pair of points $x$ and $y$ in $M$ there exists $c$ in $M$ such that $d(x,c)$ and $d(c,y)$ are both smaller than :$/\; +\; \backslash varepsilon$. 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「intrinsic 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.