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The quantitative isoperimetric inequality and related topics [PDF]

Bulletin of Mathematical Sciences, 2015
We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities.
Nicola Fusco
exaly   +2 more sources

A quantitative isoperimetric inequality on the sphere

Advances in Calculus of Variations, 2017
In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set ...
Nicola Fusco, Verena Bögelein, Frank Duzaar   +2 more
exaly   +2 more sources

A Selection Principle for the Sharp Quantitative Isoperimetric Inequality [PDF]

Archive for Rational Mechanics and Analysis, 2012
We introduce a new variational method for the study of isoperimetric inequalities with quantitative terms.The method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two notable
Gian Paolo Leonardi, Marco Cicalese
exaly   +2 more sources

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Isoperimetric Inequalities and Eigenvalues

SIAM Journal on Discrete Mathematics, 1997
Summary: An upper bound is given on the minimum distance between \(i\) subsets of same size of a regular graph in terms of the \(i\)th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the \(i\)th largest eigenvalue for any integer \(i\). Our bounds are shown to be asymptotically tight for explicit families of graphs
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ISOPERIMETRIC INEQUALITIES FOR MULTIVARIFOLDS

Mathematics of the USSR-Izvestiya, 1986
Developing the theory of multivarifolds the author establishes new isoperimetric inequalities. The main result can be stated as follows: ''Let W be a \((k+1)\)-dimensional compact Riemannian manifold with boundary \(\partial W\), and \(g: \partial W\to R^ n\) a fixed mapping of class \(C^ r\) (resp. a locally Lipschitz mapping).
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Stability of a reverse isoperimetric inequality [PDF]

Journal of Mathematical Analysis and Applications, 2009
In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303–308], which states that if K
Huiping Xu
exaly   +2 more sources

An Isoperimetric Inequality on the Discrete Torus

SIAM Journal on Discrete Mathematics, 1990
Summary: The discrete torus is the graph on \(\mathbb{Z}^ n_ k=(\mathbb{Z}/k\mathbb{Z})^ n\) in which \(x=(x_ i)^ n_ 1\) is joined to \(y=(y_ i)^ n_ 1\) if for some \(i\) there is \(x_ i=y_ i\pm1\) and \(x_ j=y_ j\) for all \(j\neq i\). For a set \(A\subset\mathbb{Z}^ n_ k\) and a natural number \(t\), let \(A_{(t)}\) be the set of vertices of ...
Béla Bollobás, Imre Leader
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An Isoperimetric Inequality for Tetrahedra

Canadian Mathematical Bulletin, 1966
Let T be a tetrahedron and let V(T) and L(T) denote its volume and the sum of its edge-lengths. In this note we prove Theorem 1. with equality if and only if the tetrahedron T is regular.
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Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

manuscripta mathematica, 1998
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The Stability of the Isoperimetric Inequality

2017
These lecture notes contain the material that I presented in two summer courses in 2013, one at the Carnegie Mellon University and the other one in a CIME school at Cetraro. The aim of both courses was to give a quick but comprehensive introduction to some recent results on the stability of the isoperimetric inequality.
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