Abstract:We study a geometric analogue of the large deviation principle for mixed measures associated with a class of $\log$-concave probability measures whose densities depend on the gauge of a convex body. For convex bodies in $\mathbb{R}^n$, we prove a geometric large-deviation-type asymptotic for first-order mixed measures, in which the decay under dilation is governed by a natural inradius associated with the measure. In the planar case, we derive an explicit representation and prove a genuine logarithmic limit for second-order mixed measures. As an application, we prove a comparison theorem showing that asymptotic dominance under dilation forces inclusion between convex bodies.
From: Artem Zvavitch [view email]
[v1]
Sat, 21 Feb 2026 18:37:35 UTC (20 KB)
[v2]
Tue, 24 Feb 2026 13:52:04 UTC (27 KB)
[v3]
Fri, 14 Aug 2026 15:16:45 UTC (22 KB)
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