Geometric Large-Deviation-Type Principles for Mixed Measures

TL;DR

This paper develops geometric large-deviation principles for mixed measures related to log-concave probability measures and convex bodies, revealing asymptotic behaviors and inclusion relations.

Contribution

It introduces large-deviation-type asymptotics for mixed measures associated with log-concave measures and convex bodies, including explicit formulas and comparison theorems.

Findings

Asymptotic decay governed by inradius for convex bodies

Explicit integral representation for planar second-order measures

Inclusion relations derived from asymptotic dominance

Abstract

We study an analogue of the large deviation principle for mixed measures associated with a class of $\log$-concave probability measures whose densities depend on the gauge function of a convex body. For convex bodies in $\mathbb{R}^n$, we prove a geometric large-deviation-type asymptotic for first-order mixed measures, where the decay under dilation is governed by a natural inradius associated with the measure. In the planar case, we derive an explicit integral representation for second-order mixed measures and obtain a corresponding asymptotic. As an application, we prove a comparison theorem showing that asymptotic dominance under dilation forces inclusion between convex bodies.