where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. The purpose of this paper is to recount some of the most interesting of the many sharpened forms, generalizations, and applications of this inequality, with emphasis on recent contributions. Earlier work is summarized in the book of Hadwiger [1], Other general references, varying from very elementary to quite technical are Kazarinoff [1], Pólya [2, Chapter X], Porter [1], and the books of Blaschke listed in the bibliography. Most books on convexity also contain a discussion of the isoperimetric inequality from that perspective. One aspect of the subject is given by Burago [1]. Others may be found in a recent paper of the author [4] on Bonnesen inequalities and in the book of Santaló [4] on integral geometry and geometric probability. An important note: we shall not go into the area of so-called \"isoperimetric problems\". Those are simply variational problems with constraints, whose name derives from the fact that inequality (1) corresponds to the first example of such a problem: maximize the area of a domain under the constraint that the length of its boundary be fixed. There are also the \"isoperimetric inequalities\" of mathematical physics. They are special cases of isoperimetric problems in which typically some physical quantity, usually represented by the eigenvalues of a differential equation, is shown to be extremal for a circular or spherical domain. Extensive discussions of such problems can be found in the book of Pólya and Szegö [1] and the review article by Payne [1]. We shall discuss them here only insofar as they relate to the main subject of this paper. What we shall concentrate on here is \"the\" isoperimetric inequality (1) and other geometric versions and generalizations of it. We shall also consider

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The isoperimetric inequality

Isoperimetric Inequalities and Their Applications

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

*The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He would like to thank the Physics Department and others at the Universidad de Chile for their hospitality during his visit in April, 1990, when much of this research was completed. The second author was supported in part by FONDECYT (Chile), projects 0132-88 and 1238-90. Both authors also thank Fritz Gesztesy for general comments and encouragement.

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions

Let Ω be a bounded smooth domain in ℝn and let 0 = λ1 ≤ λ2 ≤ … denote the eigenvalues of the Stekloff problem: Δu = 0 in Ω and (∥u)/(∥ν) = λi on ∥Ω. We show that , where denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as Ω. The proof is based on a weighted isoperimetric inequality.

An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem

Das betrachtete Eigenwertproblem kann aufgefasst werden als dasjenige einer schwingenden Membran mit teilweise festem, teilweise freiem Rand, welche aber nicht im Innern, sondern auf dem freien Randteil Massen tragt.Die Eigenfunktionen sind harmonisch: das Stekloffsche Problem mit dem zugehorigen Rayleighschen Prinzip liefert fur harmonische Funktionen andere Erkenntnisse als das Dirichletsche Problem mit dem Dirichletschen Prinzip [siehe insbesondere die Ungleichung (3)]. Isoperimetrische Ungleichungen werden durch konforme Abbildung auf ein Normalgebiet und Anwendung des Rayleighschen Prinzips auf die «verpflanzten» (siehePolya-Szego [14]) Eigenfunktionen des Normalgebietes hergeleitet.

Joseph Hersch

Papers

Extremal principles and isoperimetric inequalities for some mixed problems of Stekloff's type

On symmetric membranes and conformal radius: Some complements to Pólya's and Szegö's inequalities

On the torsion function, Green's function and conformal radius: An isoperimetric inequality of Pólya and Szegö, some extensions and applications

On harmonic measures, conformal moduli and some elementary symmetry methods

On the reflection principle and some elementary ratios of conformal radii