An Inequality for the Compositions of Convex Functions with Convolutions and an Alternative Proof of the Brunn–Minkowski–Kemperman Inequality

Takashi Satomi

Informatics and Automation, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 280-297

Résumé

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions $\phi _1,\phi _2: G\to [0,1]$ satisfy $\|\phi _1\|\leq \|\phi _2\|$ and $\|\phi _1\| + \|\phi _2\| \leq m(G)$, where $\|\cdot \|$ denotes the $L^1$-norm with respect to a Haar measure $dg$ on $G$. We have the following inequality for any convex function $f: [0,\|\phi _1\|]\to \mathbb R $ with $f(0) = 0$: $\int _{G} f \circ (\phi _1 * \phi _2)(g)\,dg \leq 2 \int _{0}^{\|\phi _1\|} f(y)\,dy + (\|\phi _2\| - \|\phi _1\|) f(\|\phi _1\|)$. As a corollary, we have a slightly stronger version of the Brunn–Minkowski–Kemperman inequality. That is, we have $\mathrm {vol}_*(B_1 B_2) \geq \mathrm {vol}(\{g\in G \mid 1_{B_1} * 1_{B_2}(g) > 0\}) \geq \mathrm {vol}(B_1) + \mathrm {vol}(B_2)$ for any non-null measurable sets $B_1,B_2 \subset G$ with $\mathrm {vol}(B_1) + \mathrm {vol}(B_2) \leq m(G)$, where $\mathrm {vol}_*$ denotes the inner measure and $1_B$ the characteristic function of $B$.

Détail

Mots-clés : convolution
Keywords: convexity, locally compact group, combinatorial inequality, geometric measure theory.

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