翻訳と辞書 Words near each other ・ Filley ・ Filley Township, Gage County, Nebraska ・ Filley, Nebraska ・ Fillgraderstiege ・ Fillibusters Camp ・ Fillie Lyckow ・ Fillies' Mile ・ Fillies' Sprint Stakes ・ Fillies' Trial Stakes ・ Filligar ・ Filling ・ Filling area conjecture ・ Filling carousel ・ Filling factor ・ Filling Factories in the United Kingdom ・ Filling radius ・ Filling station ・ Filling Station (magazine) ・ Filling station attendant ・ Filling the tree ・ Filling Up the City Skies ・ Filling Up with Heaven ・ Filling-in ・ Fillinges ・ Fillingham ・ Fillion ・ Fillip ・ Fillip Rodrigues Da Silva ・ Fillip Williams ・ Fillipe Soutto Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Filling radius ： ウィキペディア英語版 Filling radius In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling radius of a simple loop ''C'' in the plane is defined as the largest radius, ''R'' > 0, of a circle that fits inside ''C'': :$\backslash mathrm(C\backslash subset\; \backslash mathbb^2)\; =\; R.$ ==Dual definition via neighborhoods== There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the $\backslash varepsilon$-neighborhoods of the loop ''C'', denoted :$U\_\backslash varepsilon\; C\; \backslash subset\; \backslash mathbb^2.$ As $\backslash varepsilon>0$ increases, the $\backslash varepsilon$-neighborhood $U\_\backslash varepsilon\; C$ swallows up more and more of the interior of the loop. The ''last'' point to be swallowed up is precisely the center of a largest inscribed circle. Therefore we can reformulate the above definition by defining $\backslash mathrm(C\backslash subset\; \backslash mathbb^2)$ to be the infimum of $\backslash varepsilon\; >\; 0$ such that the loop ''C'' contracts to a point in $U\_\backslash varepsilon\; C$. Given a compact manifold ''X'' imbedded in, say, Euclidean space ''E'', we could define the filling radius ''relative'' to the imbedding, by minimizing the size of the neighborhood $U\_\backslash varepsilon\; X\backslash subset\; E$ in which ''X'' could be homotoped to something smaller dimensional, e.g., to a lower-dimensional polyhedron. Technically it is more convenient to work with a homological definition. 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「Filling radius」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.