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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'': : ==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 -neighborhoods of the loop ''C'', denoted : As increases, the -neighborhood 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 to be the infimum of such that the loop ''C'' contracts to a point in . 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 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
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