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Topic: Isoperimetric inequality

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Isoperimetric dimension

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	Isoperimetric Theorem
		Isoperimetric Inequality can be generalized in several ways.
		The inequality translates into two equivalent statements of which one claims that among all three dimensional solid bodies with a given surface area the sphere has the largest volume.
		(a+b)/2, a particular case of the inequality between the geometric an arithmetic means.
	www.cut-the-knot.org /do_you_know/isoperimetric.shtml (1267 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The classical isoperimetric property, discovered by Queen Dido shortly after her arrival at the coast of Africa in 900 B.C., states that amongst all figures of equal perimeter the circle encloses the largest area.
		These inequalities and subsequent extensions by Brascamp, Lieb, and Luttinger, provide a powerful and elegant method for proving not only the classical isoperimetric inequality but also many of its physical relatives such as the Rayleigh-Faber-Krahn isoperimetric inequality for the lowest eigenvalue of the Laplacian and Pólya's isoperimetric inequalities for torsional rigidity and electrostatic capacity.
		From the point of view of probability, Luttinger's inequalities are inequalities about the finite dimensional distributions of Brownian motion and raise questions concerning their validity for other stochastic processes.
	www.ipam.ucla.edu /abstract.aspx?tid=5126 (264 words)

	William Beckner: Research (Site not responding. Last check: 2007-10-29)
		Certainly the isoperimetric inequality is the cornerstone for the analysis of geometric variational problems, and can be linked to the ideas of entropy and the log Sobolev inequality through the Levy-Gromov functional.
		Sobolev inequalities, the Poisson semigroup and analysis on the sphere, Proc.
		Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann.
	www.ma.utexas.edu /users/beckner/research.html (305 words)

	Chavel Eigenvalues in Riremannian Geometry - Isaac Chavel (Site not responding. Last check: 2007-10-29)
		C here refers to some constant, which does not depend on D (it may depend on the manifold and on d).The isoperimetric dimension of M is the supremum on all d-s such that M satisfies a d-dimensional isoperimetric inequality.
		Any compact manifold has isoperimetric dimension 0.It is also possible for the isoperimetric dimension to be larger than the topological dimension.
		This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group.A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph.
	www.vikramasila.org /397013_isaac-chavel_0121706400chaveleigenvaluesinriremanniangeometryusedartbook.html (956 words)

	WWU Math Department - Colloquium
		If you were to ask a middle school student to enclose the largest area with a fixed length of string, it is more than likely that he or she would mold the string into a circle.
		While this two-dimensional application of the isoperimetric inequality has been long known, few mathematicians are aware of how this inequality is intimately related to an interwoven web of inequalities stretching through not only geometry, but also analysis.
		While the statement of the inequality is simple, and its implications are quite intuitive, many proofs of this inequality are difficult.
	www.ac.wwu.edu /~mathweb/colloquium/c_052605.htm (154 words)

	Relative Isoperimetric Inequality For Domains Outside A Convex Set - Choe (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		2: A sharp four-dimensional isoperimetric inequality (context) - Croke - 1984
		5 Isoperimetric inequalities in Riemannian manifolds (context) - Gromov - 1986
		4 An optimal relative isoperimetric inequality in concave cyli..
	citeseer.ist.psu.edu /452881.html (491 words)

	Hurwitz's proof of the the isoperimetric inequality (Site not responding. Last check: 2007-10-29)
		A fairly simple modern analytic proof of the isoperimetric inequality was given by A. Hurwirtz in 1902.
		Noting that in terms of the parametrization of C, the formulas for A follow easily.
		Hurwitz's proof also uses a famous inequality known as Wirtinger's inequality.
	www.math.jhu.edu /~js/Math427/coursenotes/node4.html (156 words)

	CiteULike: Tag inequality (Site not responding. Last check: 2007-10-29)
		An isoperimetric inequality for logarithmic capacity of polygons
		Commentary: Income inequality and reproductive outcomes--that model is best which models the least.
		Inequity and inequality in the use of health care in England: an empirical investigation
	www.citeulike.org /tag/inequality (367 words)

	Mathematische Veröffentlichungen (Site not responding. Last check: 2007-10-29)
		The idea of applying isoperimetric functions to group theory is due to M.
		If the Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub - exponential isoperimetric inequality, as well as a polynomial isodiametric inequality.
		If a finite presentation satisfies the condition V or V, then it has a linear isoperimetric* inequality and hence the group is hyperbolic.
	www.ph-karlsruhe.de /~rosebrock/web/mathe/mathindx.html (1085 words)

	The Isoperimetric inequality (Site not responding. Last check: 2007-10-29)
		She cut the hide into thin strips and tied these end to end and formed them into a semicircle, shrewdly using the straight coastline as part of the boundary.
		This isoperimetric theorem was known to the ancient Greeks although a rigorous proof did not appear until the nineteenth century.
		The isoperimetric theorem can be give a more quantitative and useful form called the isoperimetric inequality.
	www.math.jhu.edu /~js/Math427/coursenotes/node3.html (619 words)

	Fall 2005 : Topics in Probabilistic Methods for Discrete Mathematics
		Reformulation of Talagrand's inequality for "configuration functions" and its applications to Longest increasing subsequence and Longest common subsequence; Reformulation of Talagrand's inequality for "certifiable functions" (as an extension of configuration functions) with various short applications.
		Proof of the isoperimetric inequality for "certifiable functions" with an application (in random subgraphs) comparing it with IBDI; discussion of geometric interpretation of Talagrand's distance function.
		Comparison of various forms (corollaries) of the Azuma-Hoeffding Inequality applied to the expectation of the chromatic number of random graphs, and occupancy (balls and bins) problem.
	www.math.uiuc.edu /~hkaul/MethodsFall05.html (863 words)

	Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives (Cambridge Tracts in Mathematics) by ...
		This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds.
		In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds.
		The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day--and beyond!
	www.gettextbooks.com /isbn_0521802679.html (144 words)

	The Gaussian isoperimetric inequality and decoding error probabilities for the Gaussian channel (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Abstract: The Gaussian isoperimetric inequality states that among all sets in R n with prescribed Gaussian measure, the half-spaces have minimal Gaussian perimeter.
		8 and an elementary proof of the isoperimetric inequality in G..
		4 Discrete isoperimetric inequalities and the probability of a..
	citeseer.ist.psu.edu /tillich01gaussian.html (317 words)

	Bulletin of the American Mathematical Society
		Abstract: In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality.
		The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of
		J. Mecke and A. Schwella, Inequalities in the sense of Brunn-Minkowski, Vitale for random convex bodies, preprint.
	www.ams.org /bull/2002-39-03/S0273-0979-02-00941-2/home.html (1952 words)

	Isoperimetric Inequalities - Cambridge University Press (Site not responding. Last check: 2007-10-29)
		In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitative features of the inequality which provide insight into the coarse geometry at infinity of Riemannian manifolds.
		The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day - and beyond!
		First, it nicely explains the story of the classical isoperimetric inequality, a result with a big disproportion between the ease of formulation and difficulty of the proof.
	www.cup.cam.ac.uk /aus/catalogue/catalogue.asp?isbn=0521802679 (232 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Since the isoperimetric inequality for R^n equipped with Gaussian measure is a consequence of Bobkov's inequality, the result supplies the cases of equality for this isoperimetrc inequality as well.
		As in the more familiar setting of R^n with Lebesgue measure, the `sharp' form of the isoperimetric inequality on Gauss space gives rise to a `sharp' rearrangement inequality for functionals which involve an energy arising from a gradient.
		A typical example is an energy of the form \int grad u^2 d\gamma, where \gamma is the Gaussian measure on R^n.
	www.math.virginia.edu /~ji2k/Mathphys/Sep15 (122 words)

	Mudd Math Fun Facts: Isoperimetric Inequality
		All students "know" that the area enclosed by a plane curve of a given perimeter is maximized when the curve is a circle.
		The result quoted above can be presented in the following way: given a closed surface of a given area, the volume enclosed by the surface is constrained by the isoperimetric inequality.
		The proof of the inequality in three dimensions is beyond an elementary course, but it is discussed in Chapter 7 of the Courant and Robbins reference.
	www.math.hmc.edu /funfacts/ffiles/20004.2.shtml (242 words)

	Planimeters and Isoperimetric Inequalities
		The isoperimetric inequality for a region in the plane bounded by a simple closed curve states that
		There is a very simple and intuitive proof of the isoperimetric inequality based on how a planimeter works.
		A similar proof that works in spherical and hyperbolic geometry is contained in the paper Planimeters and Isoperimetric Inequalities on Constant Curvature Surfaces.
	persweb.wabash.edu /facstaff/footer/planimeter/Isoperem/pln&isop.htm (455 words)

	MAT-Report no. 2000-03.
		The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g.
		On the other hand, when this upper bound on the sectional curvatures of the ambient manifold is non-positive, the second named author has obtained an isoperimetric inequality for the extrinsic balls, see [11].
		In the present paper we extend this isoperimetric inequality to hold for extrinsic balls in minimal submanifolds of a riemannian manifold with sectional curvatures bounded from above by any constant.
	www2.mat.dtu.dk /publications/uk?id=206 (179 words)

	Atlas: A quadratic isoperimetric inequality for free-by-cyclic groups by Daniel Groves (Site not responding. Last check: 2007-10-29)
		We prove that all finitely generated free-by-cyclic groups have a quadratic isoperimetric inequality.
		This is similar to surface-by-cyclic groups, but in contrast to abelian-by-cyclic groups, two analagous classes of groups.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caja-10.
	atlas-conferences.com /c/a/j/a/10.htm (118 words)

	Isoperimetric constants for product probability measures, S. G. Bobkov, C. Houdré
		A dimension free lower bound is found for isoperimetric constants of product probability measures.
		[3] Bobkov, S. An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space.
		[15] Talagrand, M. A new isoperimetric inequality and the concentration of measure phenomenon.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aop/1024404284 (214 words)

	Project C10 - Isoperimetric Inequality
		Suppose you are given a length of rope and you tie the ends together to form a simple closed curve.
		Explain how the isoperimetric inequality justifies the following statement: Of all simple closed curves of fixed length, the circle encloses the greatest area.
		Write a brief history of the isoperimetric inequality.
	faculty.prairiestate.edu /skifowit/htdocs/projects/c10.htm (480 words)

	Journal of Inequalities and Applications (Site not responding. Last check: 2007-10-29)
		From this we derive inequalities comparing a weighted Sobolev norm of a given function with the norm of its symmetric decreasing rearrangement.
		Furthermore, we use the inequality to obtain comparison results for elliptic boundary value problems.
		Keywords and phrases: Weighted isoperimetric inequality; Weighted Sobolev norm; Symmetric decreasing rearrangement; Comparison theorem.
	www.hindawi.com /journals/jia/volume-4/S1025583499000375.html (94 words)

	A. Burchard and L. E. Thomas: "On an isoperimetric inequality ..." (Abstract) (Site not responding. Last check: 2007-10-29)
		Burchard and L. Thomas: "On an isoperimetric inequality..." (Abstract)
		"On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop", submitted May 2005.
		To our knowledge, the full conjecture remains open.
	www.math.virginia.edu /~ab4v/abstracts/loop.html (153 words)

	The world's top isoperimetry websites
		Steiner's proof was completed later by several other mathematicians.
		The isoperimetric theorem generalises to higher dimensional spaces and non-Euclidean spaces.
		The isoperimetric problem has been generalised to many different setting in advanced mathematics, and has generated new areas of current research.
	dirs.org /wiki-article-tab.cfm/isoperimetry (386 words)

	Differential Geometry Seminar, March 27, 2003 (Site not responding. Last check: 2007-10-29)
		Abstract: The classical isoperimetric inequality for a domain D in R
		It has been conjectured that this inequality should hold for any minimal submanifold M
		In this talk we consider another optimal extension of the isoperimetric inequality, called the relative isoperimetric inequality: Given a domain D outside a convex domain C in R
	www.math.umn.edu /~gulliver/abs/choe1.html (147 words)

	Problem 12: The Isoperimetric Inequality and other Geometric Applications
		Suppose that C is a closed curve in the plane such that it is of class C
		Note that since both area and perimeter must be greater than or equal to zero, it is implied that
		One implication of this famous inequality is that the geometric shape that encompasses the greatest area with respect to its perimeter is the circle, in which case
	home.wlu.edu /~mcraea/GeometricProbabilityFolder/ApplicationsConvexSets/Problem12/problem12.html (726 words)

	Amazon.ca: Books: Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives (Site not responding. Last check: 2007-10-29)
		An introduction to isoperimetric inequalities, treating isoperimetric inequalities in Euclidian space.
		Also contrasts rough inequalities in noncompact Riemannian manifolds.
		Offers a plethora of ideas and techniques of isoperimetric inequalities.
	www.amazon.ca /exec/obidos/ASIN/0521802679 (330 words)

	Journal of Inequalities and Applications (Site not responding. Last check: 2007-10-29)
		Betta, M. Brock, A. Mercaldo, and M. Posteraro: A weighted isoperimetric inequality and applications to symmetrization
		Brock, F. (with M. Betta, A. Mercaldo, and M. Posteraro): A weighted isoperimetric inequality and applications to symmetrization
		Nibbi, Roberta: Some generalized Poincaré inequalities and applications to problems arising in electromagnetism
	www.hindawi.com /journals/jia/volume-4/ai.html (646 words)

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