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Cheeger constant
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold ''M'' is a positive real number ''h''(''M'') defined in terms of the minimal area of a hypersurface that divides ''M'' into two disjoint pieces. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace-Beltrami operator on ''M'' to ''h''(''M''). This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.
== Definition ==
Let ''M'' be an ''n''-dimensional closed Riemannian manifold. Let ''V''(''A'') denote the volume of an ''n''-dimensional submanifold ''A'' and ''S''(''E'') denote the ''n''−1-dimensional volume of a submanifold ''E'' (commonly called "area" in this context). The Cheeger isoperimetric constant of ''M'' is defined to be
:
where the infimum is taken over all smooth ''n''−1-dimensional submanifolds ''E'' of ''M'' which divide it into two disjoint submanifolds ''A'' and ''B''. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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