Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. (English) Zbl 0841.49024

This paper is an exposition of some of the semigroup tools which may be used to investigate the isoperimetric inequality in Euclidean and Gauss space. Following the work of N. Varopoulos in his functional approach to isoperimetric inequalities on groups and manifolds, the author observes that the classical isoperimetric inequality in \(\mathbb{R}^n\) is equivalent to saying that the \(L^2\)-norm of the heat semigroup acting on characteristic functions of sets increases under isoperimetric rearrangement. Then, the author checks the corresponding property in Gauss space and, inspired by the approach of B. Maurey and G. Pisier to the concentration of measure phenomenon, the author surveys how the various properties of the Ornstein-Uhlenbeck semigroup such as the commutation property or hypercontractivity can yield in a simple way both the concentration of measure phenomenon and (a form of) the isoperimetric inequality for Gauss measures.


49Q20 Variational problems in a geometric measure-theoretic setting
60E15 Inequalities; stochastic orderings
28A75 Length, area, volume, other geometric measure theory
52A40 Inequalities and extremum problems involving convexity in convex geometry
47D07 Markov semigroups and applications to diffusion processes
58J35 Heat and other parabolic equation methods for PDEs on manifolds