# The isoperimetric inequality

@article{Osserman1978TheII,
title={The isoperimetric inequality},
author={Robert Osserman},
journal={Bulletin of the American Mathematical Society},
year={1978},
volume={84},
pages={1182-1238}
}
• R. Osserman
• Published 1978
• Mathematics
• Bulletin of the American Mathematical Society
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… Expand
686 Citations

#### Paper Mentions

Some Isoperimetric Inequalities and Their Applications
We denote by Sn the Euclidean sphere of R", equipped with the geodesic distance Q and a rotation invariant probability pn. For a (measurable) subset A of S,„ consider the set Au of points of SnExpand
The Standard Isoperimetric Theorem
A highly informative prototypal theorem says that among all subsets of Euclidean n-dimensional space ℝn having a prescribed measure the balls take the smallest perimeter. The present chapter isExpand
A generalization of the isoperimetric inequality in the hyperbolic plane
where the curve is given by a smooth immersion f of a circle into E and w I (p) denotes the winding number of f w.r.t, p ~ E. Equality holds if and only if the curve is a circle traversed in the sameExpand
The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces
Let (M, g) be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature K < -1. If f is a compactly supported function of bounded variation on M, then f satisfies theExpand
Extremum Problems for Convex Discs and Polyhedra
Geometric extremum problems were already investigated in ancient times, and the isoperimetric problem in particular attracted the attention of mathematicians. In the 19th century considerableExpand
Sharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane
An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to anExpand
Lower bounds on Ricci curvature and the almost rigidity of warped products
• Mathematics
• 1996
The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem,Expand
Affine Isoperimetric Inequalities for Piecewise Linear Surfaces
• Mathematics
• 2002
We consider affine analogues of the isoperimetric inequalityin the sense of piecewise linear (PL) manifolds. Given a closed polygon P having n edges, embedded in R d , we give upper and lower boundsExpand
Higher order Poincare inequalities and Minkowski-type inequalities
We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz’s proof of the Wirtinger’s inequality using Fourier theory. We then give some geometricExpand
Stability of a reverse isoperimetric inequality
• Mathematics
• 2009
Abstract 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,Expand

#### References

SHOWING 1-10 OF 223 REFERENCES
A geometrical isoperimetric inequality and applications to problems of mathematical physics
The classical isoperimetric inequality states that among all closed curves of given circumference the circle encloses the largest area. This inequality has been considerably generalized by A. D.Expand
Inequalities involving integrals of functions and their derivatives
Inequalities of the sort to be discussed in this paper are considered in Chapter 7 of Hardy, et al. [l]. Th ere these problems are considered principally from the point of view of the calculus ofExpand
Isoperimetric inequalities on curved surfaces
• Mathematics
• 1980
In this paper we extend the solutions of Lord Rayleigh’s and St. Venant’s conjectures to bounded simply connected domains on curved 2-dimensional Riemannian manifolds. The conjectures investigate theExpand
The isoperimetric theorem for curves on minimal surfaces
A short proof is given for a sharpened form of the isoperimetric inequality for curves on minimal surfaces. By following a line of development used by Sachs [7] in treating inequalities for planeExpand
On isoperimetric and various other inequalities for a manifold of bounded curvature
| I . F U N D A M E N T A L D E F I N I T I O N S A N D F O R M U L A T I O N O F R E S U L T S Le t F be a compac t two--dimensi0nal manifold of bounded cu rva tu re [1] with a nonempty edge L consExpand
Some Wirtinger-Like Inequalities
This paper extends a variational inequality [G. Hardy, J. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1967; p.182] for real valued functions and their derivatives toExpand
A Proof of a General Isoperimetric Inequality for Surfaces
• Mathematics
• 1978
(1.1) Let M be a two-dimensional C2-manifold endowed with a C2-Riemannian metric. We say that M is a generalized surface if the metric in M is allowed to degenerate at isolated points; such pointsExpand
On the first variation of a varifold
Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our mainExpand
A characteristic property of spheres
SummaryWe prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that {}_{\partialExpand
AN INEQUALITY OF THE ISOPERIMETRIC TYPE FOR A DOMAIN IN A RIEMANNIAN SPACE
We consider in the n-dimensional Riemannian space a domain with compact closure T bounded by a regular hypersurface Γ . We assume that the sectional curvatures in T are positive and the boundary Γ isExpand