On discrete Borell–Brascamp–Lieb inequalities

If f,g,h:Rn⟶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 Zn: under the same assumption as before, for A,B⊂Zn}, then ∑A+Bh≥[(∑rf(A)f)q+(∑Bg)q]1/q, where rf(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.