# On discrete Borell–Brascamp–Lieb inequalities

### David Iglesias

Universidad de Murcia, Spain### Jesús Yepes Nicolás

Universidad de Murcia, Spain

## Abstract

If $f,g,h:R_{n}⟶R_{≥0}$ 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 $−1/n≤p≤∞$, $p=0$, 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 $Z_{n}$: under the same assumption as before, for $A,B⊂Z_{n}$}, then $∑_{A+B}h≥[(∑_{rf(A)}f)_{q}+(∑_{B}g)_{q}]_{1/q}$, where $r_{f}(A)$ 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.

## Cite this article

David Iglesias, Jesús Yepes Nicolás, On discrete Borell–Brascamp–Lieb inequalities. Rev. Mat. Iberoam. 36 (2020), no. 3, pp. 711–722

DOI 10.4171/RMI/1145