# 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} }

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

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#### References

SHOWING 1-10 OF 223 REFERENCES

A geometrical isoperimetric inequality and applications to problems of mathematical physics

- Mathematics
- 1974

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

- Mathematics
- 1967

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 of… Expand

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 the… Expand

The isoperimetric theorem for curves on minimal surfaces

- Mathematics
- 1978

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 plane… Expand

On isoperimetric and various other inequalities for a manifold of bounded curvature

- Mathematics
- 1969

| 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 cons… Expand

Some Wirtinger-Like Inequalities

- Mathematics
- 1979

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 to… Expand

A Proof of a General Isoperimetric Inequality for Surfaces

- Mathematics
- 1978

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On the first variation of a varifold

- Mathematics
- 1972

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 main… Expand

A characteristic property of spheres

- Mathematics
- 1962

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
$${}_{\partial… Expand

AN INEQUALITY OF THE ISOPERIMETRIC TYPE FOR A DOMAIN IN A RIEMANNIAN SPACE

- Mathematics
- 1973

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 is… Expand