An inequality for the compositions of convex functions with convolutions and an alternative proof of the Brunn-Minkowski-Kemperman inequality

Abstract

Let $m(G)$ be the infimum of the volumes of all open subgroups of a unimodular locally compact group $G$. Suppose integrable functions ${\varphi }_1,{\varphi }_2:G\to [0,1]$ satisfy $\Vert {\varphi }_1\Vert \le \Vert {\varphi }_2\Vert$ and $\Vert {\varphi }_1\Vert +\Vert {\varphi }_2\Vert \le m(G)$, where $\Vert \cdot \Vert$ denotes the $L^1$-norm with respect to a Haar measure $dg$ on $G$. We have the following inequality for any convex function $f:[0,\Vert {\varphi }_1\Vert ]\to \mathbb{R}$ with $f(0)=0$: \begin{align*} \int_{G}^{} f \circ ( ϕ_1 * ϕ_2 ) (g) dg \leq 2 \int_{0}^{\| ϕ_1 \|} f(y) dy + ( \| ϕ_2 \| - \| ϕ_1 \| ) f( \| ϕ_1 \| ). \end{align*} As a corollary, we have a slightly stronger version of Brunn-Minkowski-Kemperman inequality. That is, we have \begin{align*} \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) \end{align*} for any non-null measurable sets $B_1,B_2\subset G$ with $\mathrm{vol}(B_1)+\mathrm{vol}(B_2)\le m(G)$, where ${\mathrm{vol}}_{\ast }$ denotes the inner measure and $1_B$ the characteristic function of $B$.

19 pages, 1 figure

References in corpus (2)

• Minimal and nearly minimal measure expansions in connected unimodular groups
• A nonabelian Brunn-Minkowski inequality