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	~~In [[Riemannian geometry]]~~, ~~the '''Cheeger isoperimetric constant'''~~ of ~~a [[compact space\|compact]] [[Riemannian manifold]] ''M''~~ is ~~a positive real number ''h''(''M'') defined~~ in ~~terms~~ of the ~~minimal [[surface area\|area]] of a [[hypersurface]] that divides ''M'' into two disjoint pieces of equal [[volume]]~~. ~~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 [[graph (mathematics)\|graph]]s~~.		Usе whole grain goods as an alternative to whіte, junk foods. A good source of health ƿroteіns аnd fibers is in whole wheat grains and whole grɑin products. The entire grain helps to keep yߋս haρpy longer helping enhance your choleѕtrerol leνels. 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	== ~~Definition ==~~

	~~Let ''M'' be an ''n''-dimensional [[closed manifold\|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

	: <~~math~~> ~~h(M)=\inf_E \frac{S(E)}{\min(V(A), V(B))},~~ <~~/math~~>

	~~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~~.

	~~== Cheeger's inequality ==~~

	~~The Cheeger constant ''h''(''M'') and~~ <~~math~~>~~\scriptstyle{\lambda_1(M)},~~<~~/math~~> ~~the smallest positive eigenvalue~~ of the ~~Laplacian on ''M''~~, are ~~related by~~ the ~~following fundamental inequality proved by Jeff Cheeger:~~

	: <~~math~~> ~~\lambda_1(M)\geq \frac{h^2(M)}{4}.~~ <~~/math~~>

	~~This inequality~~ is ~~optimal in~~ the ~~following sense:~~ for ~~any ''h'' > 0~~, ~~natural number ''k'' and ''ε'' > 0~~, ~~there exists~~ a ~~two-dimensional Riemannian manifold ''M'' with the isoperimetric constant ''h''(''M'') = ''h'' and such that the ''k''th eigenvalue of the Laplacian is within ''ε'' from~~ the ~~Cheeger bound (Buser~~, ~~1978)~~.

	~~== Buser's inequality ==~~

	~~Peter Buser proved an upper bound for~~ <~~math~~>~~\scriptstyle{\lambda_1(M)}~~<~~/math~~> ~~in terms~~ of the ~~isoperimetric constant ''h''(''M'')~~. ~~Let ''M'' be an ''n''-dimensional closed Riemannian manifold whose [[Ricci curvature]] is bounded below~~ by ~~−(''n''−1)''a''~~<~~sup~~>2<~~/sup~~>~~, where ''a'' ≥ 0~~. ~~Then~~

	: ~~<math> \lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M)~~.</~~math>~~

	== ~~See also~~ ==

	* [[Cheeger constant (graph theory)]]
	* [[Isoperimetric problem]]

	~~== References ==~~

	* Peter Buser, ''A note on the isoperimetric constant''. ~~Ann. Sci. École Norm. Sup. (4) 15 (1982)~~, no. ~~2, 213—230~~.~~{{MR\|0683635}}~~

	* Peter Buser, ~~"Über eine Ungleichung von Cheeger"~~. ~~Math~~. Z. ~~158 (1978), no~~. ~~3, 245–252. {{MR\|0478248}}~~

	* [[Jeff Cheeger]], ''A lower bound for the ~~smallest eigenvalue~~ of the ~~Laplacian''~~. ~~Problems in analysis (Papers dedicated~~ to [~~[Salomon Bochner]], 1969), pp~~.~~ 195–199~~. ~~Princeton Univ~~. ~~Press~~, ~~Princeton~~, N. J.~~, 1970 {{MR\|0402831}}~~

	<~~!-- This is not ideal, since mostly devoted to graphs --~~>
	* [[Alexander Lubotzky]], '~~'Discrete groups, expanding graphs~~ and ~~invariant measures''~~. ~~Progress in Mathematics~~, ~~vol 125, Birkhäuser Verlag, Basel, 1994~~

	[[~~Category~~:~~Riemannian geometry]~~]

173.49.213.165: /* History */

2013-10-23T15:52:11Z

History

← Older revision		Revision as of 17:52, 23 October 2013
Line 1:		Line 1:
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Try exchanging those meals with far healthier snack fooԁ items lіke new veցetables and reduced-fat popcorn.<br><br>Remain wholesome by applying the following tips to your lifestyle. If yоu consume the appropriatе nutrients and vitamins, ~~you can expect to appearance and fеel much healthier.~~		In [[Riemannian geometry]], the '''Cheeger isoperimetric constant''' of a [[compact space\|compact]] [[Riemannian manifold]] ''M'' is a positive real number ''h''(''M'') defined in terms of the minimal [[surface area\|area]] of a [[hypersurface]] that divides ''M'' into two disjoint pieces of equal [[volume]]. 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 [[graph (mathematics)\|graph]]s.

			== Definition ==

			Let ''M'' be an ''n''-dimensional [[closed manifold\|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

			: <math> h(M)=\inf_E \frac{S(E)}{\min(V(A), V(B))}, </math>

			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.

			== Cheeger's inequality ==

			The Cheeger constant ''h''(''M'') and <math>\scriptstyle{\lambda_1(M)},</math> the smallest positive eigenvalue of the Laplacian on ''M'', are related by the following fundamental inequality proved by Jeff Cheeger:

			: <math> \lambda_1(M)\geq \frac{h^2(M)}{4}. </math>

			This inequality is optimal in the following sense: for any ''h'' > 0, natural number ''k'' and ''ε'' > 0, there exists a two-dimensional Riemannian manifold ''M'' with the isoperimetric constant ''h''(''M'') = ''h'' and such that the ''k''th eigenvalue of the Laplacian is within ''ε'' from the Cheeger bound (Buser, 1978).

			== Buser's inequality ==

			Peter Buser proved an upper bound for <math>\scriptstyle{\lambda_1(M)}</math> in terms of the isoperimetric constant ''h''(''M''). Let ''M'' be an ''n''-dimensional closed Riemannian manifold whose [[Ricci curvature]] is bounded below by −(''n''−1)''a''<sup>2</sup>, where ''a'' ≥ 0. Then

			: <math> \lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M).</math>

			== See also ==

			* [[Cheeger constant (graph theory)]]
			* [[Isoperimetric problem]]

			== References ==

			* Peter Buser, ''A note on the isoperimetric constant''. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213—230.{{MR\|0683635}}

			* Peter Buser, "Über eine Ungleichung von Cheeger". Math. Z. 158 (1978), no. 3, 245–252. {{MR\|0478248}}

			* [[Jeff Cheeger]], ''A lower bound for the smallest eigenvalue of the Laplacian''. Problems in analysis (Papers dedicated to [[Salomon Bochner]], 1969), pp. 195–199. Princeton Univ. Press, Princeton, N. J., 1970 {{MR\|0402831}}

			<!-- This is not ideal, since mostly devoted to graphs -->
			* [[Alexander Lubotzky]], ''Discrete groups, expanding graphs and invariant measures''. Progress in Mathematics, vol 125, Birkhäuser Verlag, Basel, 1994

			[[Category:Riemannian geometry]]

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