On discrete Borell–Brascamp–Lieb inequalities

Abstract

If $f, g, h\colon \mathbb{R}^n\longrightarrow\mathbb{R}_{\geq 0}$ are non-negative measurable functions such that $h(x+y)$ is greater than or equal to the $p$-sum of $f(x)$ and $g(y)$, where $-1/n\leq p\leq\infty$, $p\neq0$, then the Borell–Brascamp–Lieb inequality asserts that the integral of $h$ is not smaller than the $q$-sum of the integrals of $f$ and $g$, for $q=p/(np+1)$.

In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice $\mathbb Z^n$: under the same assumption as before, for $A,B\subset\mathbb{Z}^n$}, then $\sum_{A+B}h\geq[(\sum_{\mathrm {r}{f}(A)} f)^q+(\sum_B g)^q]^{1/q}$, where $\mathrm{r}_{\!f}(A)$ is obtained by removing points from $A$ in a particular way, and depending on $f$. We also prove that the classical Borell–Brascamp–Lieb inequality for Riemann integrable functions can be obtained as a consequence of this new discrete version.