An isoperimetric lemma

Isoperimetric inequalities and variations on Schwarz's Lemma

In this note we prove a version of the classical Schwarz lemma for the first eigenvalue of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne and Rayner, which we reinterpret as an isoperimetric inequality for a (singular) conformal metric on a bounded domain in the plane.

Isoperimetric Problems for Convex Bodies and a Localization Lemma*

We study the smallest number qJ(K) such that a given convex body K in R n can be cut into two parts K 1 and K 2 by a surface with an (n 1)-dimensional measure ~0(K)vol(K1)vol(K2)/vol(K). Let MI(K) be the average distance of a point of K from its center of gravity. We prove for the "isoperimetric coefficient" that In 2 ~0(K) >_ M1(K) ' and give other upper and lower bounds. We conjecture that ou...

An Asymptotic Isoperimetric Inequality

For a finite metric space V with a metric ρ, let V n be the metric space in which the distance between (a1, . . . , an) and (b1, . . . , bn) is the sum ∑n i=1 ρ(ai, bi). We obtain an asymptotic formula for the logarithm of the maximum possible number of points in V n of distance at least d from a set of half the points of V , when n tends to infinity and d satisfies d √ n. 1 The Main Results Le...

An isoperimetric le

Bialostocki, A. and P. Dierker, An isoperimetric lemma, Discrete Mathematics 94 (1991) 221-228. A continuous version of the following problem is solved: Let G be a multipartite graph with a given partition of its vertex set, A, U A, U U A,,,. Find the maximum possible number of edges in G such that G has no connected component with more then t vertices.

An isoperimetric comparison theorem