Sharp estimates for the Cram\'{e}r transform of log-concave measures and geometric applications

TL;DR

This paper develops sharp estimates for the Cramér transform of log-concave measures, linking it to geometric properties like half-space depth, and applies these results to phase transitions in random polytopes and exponential separability constants.

Contribution

It introduces new comparison techniques between the Legendre transform and half-space depth for log-concave distributions, extending to multidimensional cases and deriving applications in geometric probability.

Findings

New comparison between Legendre transform and half-space depth for log-concave measures

Sharp estimates for Cramér transform of rotationally invariant measures

Results on phase transitions in expected measures of random polytopes

Abstract

We establish a new comparison between the Legendre transform of the cumulant generating function and the half-space depth of an arbitrary log-concave probability distribution on the real line, that carries on to the multidimensional setting. Combined with sharp estimates for the Cram\'{e}r transform of rotationally invariant measures, we are led to some new phase-transition type results for the asymptotics of the expected measure of random polytopes. As a byproduct of our analysis, we address a question on the sharp exponential separability constant for log-concave distributions, in the symmetric case.