On discrete Borell–Brascamp–Lieb inequalities

  • David Iglesias

    Universidad de Murcia, Spain
  • Jesús Yepes Nicolás

    Universidad de Murcia, Spain
On discrete Borell–Brascamp–Lieb inequalities cover
Download PDF

A subscription is required to access this article.


If f,g,h ⁣:RnR0f, g, h\colon \mathbb{R}^n\longrightarrow\mathbb{R}_{\geq 0} are non-negative measurable functions such that h(x+y)h(x+y) is greater than or equal to the pp-sum of f(x)f(x) and g(y)g(y), where 1/np-1/n\leq p\leq\infty, p0p\neq0, then the Borell–Brascamp–Lieb inequality asserts that the integral of hh is not smaller than the qq-sum of the integrals of ff and gg, for q=p/(np+1)q=p/(np+1).

In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice Zn\mathbb Z^n: under the same assumption as before, for A,BZnA,B\subset\mathbb{Z}^n}, then A+Bh[(rf(A)f)q+(Bg)q]1/q\sum_{A+B}h\geq[(\sum_{\mathrm {r}{f}(A)} f)^q+(\sum_B g)^q]^{1/q}, where r ⁣f(A)\mathrm{r}_{\!f}(A) is obtained by removing points from AA in a particular way, and depending on ff. 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