On discrete Borell–Brascamp–Lieb inequalities

Abstract

If $f,g,h:{\mathbb{R}}^n⟶{\mathbb{R}}_{\ge 0}$ 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 $-1/n\le p\le \infty$, $p\ne 0$, 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 ${\sum }_{A+B}h\ge [({\sum }_{\mathrm{r}f(A)}f{)}^q+({\sum }_{B}g{)}^q{]}^{1/q}$, 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.