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

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 $B$.