Geometric Large-Deviation-Type Principles for Mixed Measures

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.