Spectral Gap for log-Concave Probability Measures on the Real Line

Abstract

We propose a simple parameter to describe the exact order of the Poincaré constant (or the inverse of the spectral gap) for a log-concave probability measure on the real line. This parameter is the square of the mean value of the distance to the median. Bobkov recently derived a similar result in terms of the variance of the measure. His approach was based on the study of the Cheeger constant. Our viewpoint is quite different and makes use of the Muckenhoupt functional and of a variational computation in the set of convex functions.

Pierre Fougères: Address during the work: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France