This is an announcement for the paper "On the equivalence of modes of convergence for log-concave measures" by Elizabeth S. Meckes and Mark W. Meckes.

Abstract: An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note, we explore the extent to which different notions of distance between probability measures are comparable for log-concave distributions. Our results imply that weak convergence of isotropic log-concave distributions is equivalent to convergence in total variation, and is further equivalent to convergence in relative entropy when the limit measure is Gaussian.

Archive classification: math.PR math.FA

Submitted from: mark.meckes@case.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1312.3094

or

http://arXiv.org/abs/1312.3094