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

Silouanos Brazitikos; Giorgos Chasapis

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arXiv · 2405.07253

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

Silouanos Brazitikos·Giorgos Chasapis

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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.

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BibTeX RIS

Silouanos Brazitikos, Giorgos Chasapis. 2024-05-12. Sharp estimates for the Cram\'{e}r transform of log-concave measures and geometric applications. https://arxiv.org/abs/2405.07253

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