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

/pdf/the-isoperimetric-inequality-546lmzkbf9.pdf

The isoperimetric inequality

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We present a detailed analysis of four of the most widely used radio source-finding packages in radio astronomy, and a program being developed for the Australian Square Kilometer Array Pathfinder telescope. The four packages: sextractor, sfind, imsad and selavy are shown to produce source catalogues with high completeness and reliability. In this paper we analyse the small fraction (∼1 per cent) of cases in which these packages do not perform well. This small fraction of sources will be of concern for the next generation of radio surveys which will produce many thousands of sources on a daily basis, in particular for blind radio transients surveys. From our analysis we identify the ways in which the underlying source-finding algorithms fail. We demonstrate a new source-finding algorithm aegean, based on the application of a Laplacian kernel, which can avoid these problems and can produce complete and reliable source catalogues for the next generation of radio surveys.

/pdf/compact-continuum-source-finding-for-next-generation-radio-2ab40d08c0.pdf

Compact continuum source finding for next generation radio surveys

ROBERT OSSERMAN wrote a Ph.D. thesis on Riemann surfaces under the direction of Lars V. Ahlfors at Harvard University. He gradually moved from geometric function theory to minimal surfaces, differential geometry, isoperimetric inequalities, and some aspects of partial differential equations and ergodic theory. He has been at Stanford University since 1955 and will be spending half time as Deputy Director of MSRI in Berkeley for a period of three years, starting September 1, 1990.

Curvature in the eighties

Several computer vision algorithms rely on detecting a compact but representative set of interest regions and their associated descriptors from input data. When the input is in the form of an unorganized 3D point cloud, current practice is to compute shape descriptors either exhaustively or at randomly chosen locations using one or more preset neighborhood sizes. Such a strategy ignores the relative variation in the spatial extent of geometric structures and also risks introducing redundancy in the representation. This paper pursues multi-scale operators on point clouds that allow detection of interest regions whose locations as well as spatial extent are completely data-driven. The approach distinguishes itself from related work by operating directly in the input 3D space without assuming an available polygon mesh or resorting to an intermediate global 2D parameterization. Results are shown to demonstrate the utility and robustness of the proposed method.

/pdf/multi-scale-interest-regions-from-unorganized-point-clouds-2yn47omt65.pdf

Multi-scale interest regions from unorganized point clouds

Geometric applications of the solvability of Neumann problems on a Riemannian manifold

(1982). Mean Curvature, the Laplacian, and Soap Bubbles. The American Mathematical Monthly: Vol. 89, No. 3, pp. 180-198.

Robert C. Reilly

Papers

Geometric applications of the solvability of Neumann problems on a Riemannian manifold

Mean Curvature, the Laplacian, and Soap Bubbles