In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the...

against those of the Euclidean space). In the Euclidean space, the isoperimetric inequality says that of all bodies with the same volume, the ball has the...

In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states...

proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which...

The Pólya–Szegő inequality can be proved by combining the coarea formula, Hölder’s inequality and the classical isoperimetric inequality. If the function...

inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality. Let K and L be two n-dimensional convex bodies in n-dimensional...

perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean...

matrix. The Fisher information matrix plays a role in an inequality like the isoperimetric inequality. Of all probability distributions with a given entropy...

-sphere of radius 1. The hyperbolic space also satisfies a linear isoperimetric inequality, that is there exists a constant i {\displaystyle i} such that...

of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally...

of the length. Mikhail Gromov once voiced the opinion that the isoperimetric inequality was known already to the Ancient Greeks. The mythological tale...

Hitchin–Thorpe inequality Isoperimetric inequality Jordan's inequality Jung's theorem Loewner's torus inequality Łojasiewicz inequality Loomis–Whitney inequality Melchior's...

dimension. Loomis, L. H.; Whitney, H. (1949). "An inequality related to the isoperimetric inequality". Bulletin of the American Mathematical Society. 55...

linear isoperimetric inequality; it turns out that having such an isoperimetric inequality characterises Gromov-hyperbolic spaces. Linear isoperimetric inequalities...

σn coincides with (normalized) Haar measure on Sn. There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see...

similarity transformations of the curve. According to the isoperimetric inequality, the isoperimetric ratio has its minimum value, 4π, for a circle; any other...

integer lattice. Isoperimetric inequality Milman's reverse Brunn–Minkowski inequality Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random...

circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetric inequality. More precisely, consider a planar simple closed curve of length...

an application of the isoperimetric inequality to the function's level sets. In one dimension, this is Wirtinger's inequality for functions. However...

Bobkov's inequality is a functional isoperimetric inequality for the canonical Gaussian measure. It generalizes the Gaussian isoperimetric inequality. The...

The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants. Wirtinger's inequality also generalizes...

circle to a problem in the calculus of variations, namely the isoperimetric inequality. If a circle of radius r is centred at the vertex of an angle,...

theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved...

More generally, the Faber–Krahn inequality holds in any Riemannian manifold in which the isoperimetric inequality holds. In particular, according to...

which has a weak solution but no strong solution. The Gaussian isoperimetric inequality (proved by Vladimir Sudakov and Tsirelson, and independently by...

volume ratios and isoperimetric quotients for convex sets in and. There is also a geometric version of the more general inequality in which the maps B...

p} are related by the isoperimetric inequality p 2 > 12 3 T . {\displaystyle p^{2}>12{\sqrt {3}}T.} This is a strict inequality for isosceles triangles...

it satisfies a linear isoperimetric inequality. Moreover, there is an isoperimetric gap in the possible spectrum of isoperimetric functions for finitely...

bound on T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles: T ≤ 3 36 ( a + b + c ) 2 = 3 9 s 2...

first example goes back to Paul Lévy. According to the spherical isoperimetric inequality, among all subsets A {\displaystyle A} of the sphere S n {\displaystyle...