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Topic: Isoperimetric theorem

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Circle

Unit circle

Secant line

Euclidean geometry

	PlanetMath: isoperimetric inequality
		The advantage of this formulation is that it does not depend on the notion of surface area, and so can be generalized to arbitrary measure spaces with a metric.
		An example when this general formulation proves useful is the Talagrand's isoperimetric theory dealing with Hamming-like distances in product spaces.
		This is version 9 of isoperimetric inequality, born on 2003-10-17, modified 2005-09-17.
	planetmath.org /encyclopedia/IsoperimetricInequality.html (341 words)

	Groups with Word Problem in NP, and Higman Embeddings
		So perhaps the class of groups with word problem in NP (which by Theorem 14 is the class of all subgroups of finitely presented groups with polynomial Dehn functions) can be considered as the class of ``tame" groups.
		Indeed, Theorem 3 shows that an isoperimetric function of a group H containing a given group Gcannot be smaller than the non-deterministic time complexity T(n) of the word problem for G, and Theorem 14 shows that G can be embedded into a finitely presented group with Dehn function at most
		Theorem 16 Every countable group with solvable power and order problems is embeddable into a finitely presented group with solvable power, order and conjugacy problems.
	www.math.vanderbilt.edu /~msapir/Talk1/node6.html (1879 words)

	Isoperimetric Theorem
		Isoperimetric Theorem has been known from the time of antiquity.
		Mathematical theorems in general have premises and conclusions and assert that the latter follow from the former.
		The circle is perfect and, thus, is a good candidate to satisfy the Isoperimetric Theorem.
	www.cut-the-knot.org /do_you_know/isoperimetric.shtml (1275 words)

	Isoperimetry - Wikipedia, the free encyclopedia
		The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane.
		Modern formulations of isoperimetric problems are sometimes given in terms of sub-Riemannian geometry; Dido's problem specifically finds expression in terms of the Heisenberg group: given an arc connecting two points, the "height" z of a point in the Heisenberg group corresponds to the area subtended by the arc.
		The isoperimetric theorem generalises to higher dimensional spaces: the domain with volume 1 with the minimal surface area is always a ball.
	en.wikipedia.org /wiki/Isoperimetric_theorem (598 words)

	CBMS BRIDSON ABSTRACTS
		The word problem as measured by Dehn functions; connection to isoperimetric properties of manifolds via the Filling Theorem; the isoperimetric spectrum; the clear demarkation of hyperbolic groups and non-positively curved groups (again).
		The contrasting theorems of Bridson in the combable case.
		Theorems establishing the equivalence of hyperbolic and atoroidal properties in various contexts (3-manifolds, analytic manifolds, new results in 2003--2004).
	www.albany.edu /~ted/abstracts.html (712 words)

	Circle (Site not responding. Last check: 2007-10-10)
		Every triangle gives rise to several circles: its circumcircle containing all three vertices, its incircle lying inside the triangle and touching all three sides, the three excircles lying outside the triangle and touching one side and the extensions of the other two, and its nine point circle which contains various important points of the triangle.
		Thales' theorem states that if the three vertices of a triangle lie on a given Circle with one side of the triangle being a diameter of the circle, then the angle opposite to that side is a right angle.
		Clifford discovered, in the ordinary Euclidean plane, a "sequence or Chain of theorems" of increasing complexity, each building on the last in a natural progression.
	circle.iqnaut.net (1110 words)

	52: Convex and discrete geometry
		Loewner's theorem: there is a unique minimal-volume ellipsoid containing any given bounded set in R^n
		Quick proof of the isoperimetric inequality (that other closed curves enclose less area than a circle of the same length).
		Isoperimetric inequality (circle has maximum area for fixed perimeter): definitions, pointers, proofs
	www.math.niu.edu /~rusin/known-math/index/52-XX.html (487 words)

	Soap bubbles and isoperimetric problems
		The isoperimetric problem, in an n-dimensional Riemannian manifold, is to enclose a region of a given (n-dimensional) volume v using a hypersurface of the smallest possible "area" (n-1 dimensional volume).
		This conjecture, or now theorem, asserts that the least-area enclosure of two prescribed volumes in R^3 is the "standard double bubble", consisting of three pieces of spheres meeting along a circle at 120 degree angles.
		Suppose you try to solve the isoperimetric problem or the soap bubble problem in a product of compact Riemannian manifolds M x N, where the metric on N is rescaled to be very small with respect to M and the volumes that you want to enclose.
	math.berkeley.edu /~hutching/pub/bubbles.html (1062 words)

	Isoperimetry and concentration of measure (L24) (Site not responding. Last check: 2007-10-10)
		Each of these theorems requires its own technique, and we shall establish all the results that we shall need on the way (the Prékopa-Leindler inequality, the Brunn-Minkowski inequality, the Lévy projection theorem,...).
		We shall also investigate the many consequences of the isoperimetric theorems, and particularly those that relate to the geometry of Banach spaces (Dvoretzky's Theorem on spherical sections, Gluskin's theorem).
		A typical application of the isoperimetric theorems is that in high dimensions, a Lipschitz function takes values near its median with high probability, and the probability of large deviations is small.
	www.maths.cam.ac.uk /CASM/courses/descriptions/node10.html (268 words)

	[No title]
		January 1, 2004 Limor Ben-Efraim, Frankl and Furedi's proof of Harper's isoperimetric theorem for the discrete cube.
		March 4, 2004 an Romik, The geometry of the Sierpinski gasket Abstract: This will be an informal talk, in which I will survey some fundamental definitions and results on self-similar fractals, concentrating on the Sierpinski gasket as a well-studied example.
		Examples will include Minkowski's Theorem on the uniqueness of a star-body with given volumes of sections, and generalizations of Meyer and Pajor's results on the extremal sections of l_p balls.
	www.wisdom.weizmann.ac.il /~gideon/seminar2003_2004.txt (745 words)

	Formalizing 100 Theorems
		There exists a "top 100" of mathematical theorems on the web, which is a rather arbitrary list (and most of the theorems seem rather elementary), but still is nice to look at.
		It just shows formalizations in systems that have formalized a significant number of theorems, or that have formalized a theorem that none of the others have done.
		Theorems in the list which have not been formalized yet are in italics.
	www.cs.ru.nl /~freek/100 (349 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		The Isoperimetric Theorem Ann E. Watkins Activities to aid in the discovery that for a given perimeter the circle encloses the greatest area.
		Some Theorems Involving the Lengths of Segments in a Triangle Donald R. Byrkit and Timothy L. Dixon Proof of a theorem concerning the length of an internal angle bisector in a triangle.
		The Isoperimetric Theorem Ann E. Watkins Activities to aid in the discovery of the fact that for a given perimeter the circle encloses the greatest area.
	www.mathforum.org /mathed/mtbib/whole.text.txt (18350 words)

	Lecture Notes on Differential Geometry (Site not responding. Last check: 2007-10-10)
		Proofs of the inverse function theorem and the rank theorem.
		Regular values, proof of fundamental theorem of algebra, Smooth manifolds with boundary, Sard's theorem, and proof of Brouwer's fixed point theorem.
		Riemannian connections, brackets, proof of the fundamental theorem of Riemannian geometry, induced connection on Riemannian submanifolds, reparameterizations and speed of geodesics, geodesics of the Poincare's upper half plane.
	www.math.gatech.edu /~ghomi/LectureNotes/index.html (408 words)

	HOW ARE THINGS SHAPING UP?
		Students will extend the lines so as to make a rectangle about each triangle, and then use the formula for area of a triangle to find the areas of each of these.
		Students will use the Pythagorean Theorem to help find the perimter of each triangle.
		They will learn to use the Pythagorean Theorem to find the slant height given the height and length of the base.
	mtl.math.uiuc.edu /users/estaude/Teacher_Page/Shaping_Up.html (3558 words)

	A Sampling from a Geometry Collection (Site not responding. Last check: 2007-10-10)
		The well known Isoperimetric Theorem deals with the shapes that have the same perimeter.
		We thus obtain a family of isoperimetric shapes whose boundary consists of circular arcs and straight line segments, the derivation being based on the well known, but mostly misrepresented, identity (1).
		I have little to add here, except perhaps, that on one occasion I showed the applet and explained the theorem to two brothers, seven and ten years old, Both showed a grasp for the inductive step right away.
	www.maa.org /editorial/knot/GeometrySampling.html (1482 words)

	Proceedings of the American Mathematical Society
		We also study manifolds satisfying Hardy's inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold.
		I. Benjamini, and J. Cao, A new isoperimetric comparison theorem for surfaces of variable curvature, Duke Math.
		S.T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann.
	e-math.ams.org /proc/1999-127-09/S0002-9939-99-04849-2/home.html (346 words)

	The Isoperimetric Inequality - September 2002
		He is particularly fascinated by isoperimetric inequalities for eigenvalue problems in pde..
		A new proof (due to X Cabre) of the classical isoperimetric theorem, based on Alexandrov's id- ea of moving planes, will be presented.
		Compared to the usual proofs, which use geometric measure theory, this proof will be based on elementary ideas from calculus and partial differential equations (Laplace equation).
	www.ias.ac.in /resonance/Sept2002/Sept2002p8-18.htm (82 words)

	2005 Fall Meeting MD-DC-VA MAA
		She is editor of the Student Research Projects for the College Math Journal and editor of the Pi Mu Epsilon Journal.
		The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area.
		The isoperimetric theorem then states that the curve is a circle.
	www.morgan.edu /maa/fall05/default.html (1691 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Pavel Exner, Evans M. Harrell, and Michael Loss Inequalities for means of chords, with application to isoperimetric problems (29K, LaTeX) ABSTRACT.
		We consider a pair of isoperimetric problems arising in physics.
		We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $\Gamma$ is a circle.
	www.ma.utexas.edu /mp_arc/a/05-300 (158 words)

	Motivate : Mathematics Videoconferences for Schools.
		This is such an important result that it has a special name.
		It is called the isoperimetric theorem, from the Greek words for 'same' and 'perimeter'.
		Task 1 should have given you a practical demonstration that the isoperimetric theorem is true - but, of course, this is not the same as a proof.
	www.motivate.maths.org /conferences/conf27/c27_talk1.shtml (364 words)

	Fremlin --- Measure Theory (Site not responding. Last check: 2007-10-10)
		The divergence of a vector field; sets with locally finite perimeter, perimeter measures and outward-normal functions; the reduced boundary; invariance under isometries; isoperimetric inequalities; Federer exterior normals; the Compactness Theorem.
		Essential interior, closure and boundary; the reduced boundary; perimeter measures; characterizing sets with locally finite perimeter; the Divergence Theorem; calculating perimeters from cross-sectional counts; Cauchy's Perimeter Theorem; the Isoperimetric Theorem for convex sets.
		; the Isoperimetric Theorem; concentration of measure on spheres.
	www.essex.ac.uk /maths/staff/fremlin/cont47.htm (126 words)

	publications.html
		We give a very general isoperimetric comparison theorem, which implies for example that geodesic spheres in the Schwarzschild space minimize area for given volume, which in turn has applications to the Penrose Inequality in general relativity.
		We add to the literature the well-known fact that an isoperimetric hypersurfaces S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold.
		In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
	www.williams.edu /Mathematics/fmorgan/publications.html (5315 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		History of the theorem The theorem was already known 900 years BC.
		This is not a real mathematical proof, but elaboration of this example adequately illustrates the theorem.
		The circle’s perimeter is smaller that that of a rectangle having equal area.
	pumas.jpl.nasa.gov /MSWord_Examples/01_22_03_1.doc (788 words)

	The isoperimetric theorem
		Of all plane figures with a given perimeter, the circle has the greatest area.
		Alternatively, the theorem can be stated, Of all plane figures with a given area, the circle has the least perimeter.
		Elementary proofs can be found in many books under the heading "the isoperimetric theorem." It is sometimes also referred to as "Dido's problem" inspired by the legend of the founding of Carthage.
	mathcentral.uregina.ca /qq/database/QQ.09.99/bobal1.html (134 words)

	Mathematics Colloquium March 8, 2001 (Site not responding. Last check: 2007-10-10)
		Abstract: Geometrical inequalities such as the isoperimetric theorem, Brunn-Minkowski inequality and other relations between mixed-volumes are based on domination of the geometric by the arithmetic mean.
		This intuition can be made precise by exploiting the theory of optimal mappings.
		After related developments are surveyed, the method is applied to derive a new interpolation theorem extending various Euclidean inequalities to the Riemannian setting.
	www.math.virginia.edu /~colloq/2000-01/01-03-08-mccann.html (71 words)

	École d'été Surfaces Minimales et Problèmes Variationnels
		In 1884, Schwarz proved the classical isoperimetric theorem that a round ball provides the least-perimeter way to enclose given volume in R3.
		Since double bubbles have singular curves, the proof requires geometric measure theory, which uses measure theory to generalize differential geometry to singular surfaces.
		The course will include an introduction to geometric measure theory, a discussion of isoperimetric problems (including double bubble problems and spaces with densities), and open questions.
	www.math.jussieu.fr /minimal/prog_bil.html (836 words)

	Isoperimetric theorem and its variants
		Among all plane regions with a given perimeter a circle has the largest area.
		For the sake of the current discussion I'll accept Isoperimetric Theorem as a known fact on which it's easy to base a solution to the original problem.
		Assume a string is attached to the end points of the graph y=sin(x) where -1
	www.cut-the-knot.org /Generalization/isop.shtml (208 words)

	Jianguo Cao's Home Page (Site not responding. Last check: 2007-10-10)
		``A new isoperimetric comparison theorem for surfaces of variable curvature." Duke Math.
		` A new proof of the Takeuchi theorem " (with M. Shaw), Lecture Notes of Seminario Interdisplinare di Mathematica, vol.
		``An intrinsic proof of Gromoll-Grove diameter rigidity theorem" (with Hongyan Tang), accepted for publication in ``Communication in Contemporary Math." For the PDF file of the updated version (March 2006), see Cao-Tang.pdf.
	www.nd.edu /~jcao (795 words)

	[No title]
		Mathematicians were not immune, and at a mathematics conference in July, 1999, Paul and Jack Abad presented their list of "The Hundred Greatest Theorems." Their ranking is based on the following criteria: "the place the theorem holds in the literature, the quality of the proof, and the unexpectedness of the result."
		The list is of course as arbitrary as the movie and book list, but the theorems here are all certainly worthy results.
		I hope to over time include links to the proofs of them all; for now, you'll have to content yourself with the list itself and the biographies of the principals.
	personal.stevens.edu /~nkahl/Top100Theorems.html (217 words)

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