Sharp estimates for the Cram\'{e}r transform of log-concave measures and geometric applications
Probability
2025-06-17 v2 Functional Analysis
Metric Geometry
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.
Cite
@article{arxiv.2405.07253,
title = {Sharp estimates for the Cram\'{e}r transform of log-concave measures and geometric applications},
author = {Silouanos Brazitikos and Giorgos Chasapis},
journal= {arXiv preprint arXiv:2405.07253},
year = {2025}
}