Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

By Vincent Bayle, César Rosales. 2003

47 citations recorded by the source corpus.

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M n+1 , g) satisfies a second-order differential inequality that only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Q. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality, we obtain sharp comparison theorems: not only can we derive an inequality that should be compared with Levy-Gromov Inequality but we also show that if Ric > n6 on Q, then the profile of Q is bounded from above by the profile of the half-space H n+1 δ in the simply connected space form with constant sectional curvature δ. As a consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of Ω, and for the first non-zero Neumann eigenvalue for the Laplace operator on Ω.

Subfields: Applied Mathematics, Genetics

View Paper: https://www.semanticscholar.org/paper/7993e4b666003952b5cee69d49ad1dd61bef61f3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Genetics

Emanuel Milman

Citations in the corpus

Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, background): “[34, 11]) that P9(A,) is lower semicontinuous with respect to convergence of sets in the L(1)loc(9) := L 1 loc(,9volg) topology, i.”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “It is then a (tedious) exercise to follow the proof of Sternberg and Zumbrun [82] and Bayle [12] (see also [13]) and to deduce the following extension of Theorem A.”
Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound (2009, methodology): “This is also in agreement with the stronger result known in this case [29, 8], stating that in fact I n n−1 is concave.”