On $L_p$ Brunn-Minkowski type inequalities for a general class of functionals

TL;DR

The paper develops an $L_p$ Brunn-Minkowski theory for a broad class of functionals on subsets of $\\mathbb{R}^n$, with a unifying transfer principle that converts a BM inequality for the Minkowski sum into an $L_p$-type inequality for the corresponding $p$-sum. This framework is then applied to general absolutely continuous measures with radially decreasing densities, yielding an $L_p$ BM inequality for measures $\\nu$ with an admissible pair $(\\nu,\\alpha)$ and, in particular, to the Gaussian measure with $\\alpha=1/n$, including equality characterizations. The authors also derive an $L_p$ BM inequality for the Wills functional and its generalized versions, showing that these geometric functionals satisfy the same structural inequalities under $p$-sum additions. Overall, the work broadens the $L_p$ Brunn-Minkowski theory to encompass Gaussian and other radially decreasing density measures and the Wills-type functionals, providing sharp equality conditions and a versatile, model-agnostic approach.

Abstract

In this work, the $L_p$ version (for $p> 1$) of the dimensional Brunn-Minkowski inequality for the standard Gaussian measure $γ_n(\cdot)$ on $\mathbb{R}^n$ is shown. More precisely, we prove that for any $0$-symmetric convex sets with nonempty interior, any $p>1$, and every $λ\in (0,1)$, \[ γ_n\bigl((1-λ)\cdot K+_p λ\cdot L\bigr)^{p/n} \geqslant (1-λ) γ_n(K)^{p/n} + λγ_n(L)^{p/n}, \] with equality, for some $λ\in (0,1)$ and $p>1$, if and only if $K=L$. This result, recently established without the equality conditions by Hosle, Kolesnikov and Livshyts, by using a different and functional approach, turns out to be the $L_p$ extension of a celebrated result for the Minkowski sum (that is, for $p=1$) by Eskenazis and Moschidis (2021) on a problem by Gardner and Zvavitch (2010). Moreover, an $L_p$ Brunn-Minkowski type inequality is obtained for the classical Wills functional $\mathcal{W}(\cdot)$ of convex bodies. These results are derived as a consequence of a more general approach, which provides us with other remarkable examples of functionals satisfying $L_p$ Brunn-Minkowski type inequalities, such as different absolutely continuous measures with radially decreasing densities.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 1.1
• Theorem 3
• Theorem 1.2
• Definition 2.1
• Lemma 1
• Theorem 2.1
• Remark 2.1
• proof