If are non-negative measurable functions such that is greater than or equal to the -sum of and , where , , then the Borell–Brascamp–Lieb inequality asserts that the integral of is not smaller than the -sum of the integrals of and , for .
In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice : under the same assumption as before, for }, then , where is obtained by removing points from in a particular way, and depending on . 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. 711–722DOI 10.4171/RMI/1145