Restricted Prékopa-leindler Inequality

We prove a functional version of the Brunn-Minkowski inequality for restricted sums obtained by Szarek and Voicu-lescu. We only consider Lebesgue-measurable subsets of R n , and for A ⊂ R n , we denote its volume by |A|. If A, B ⊂ R n , their Minkowski sum is defined by A + B = {x + y, (x, y) ∈ A × B}. The classical Brunn-Minkowski inequality provides a lower bound for its volume. In their study of the free analogue of the entropy power inequality [SV], Szarek and Voiculescu define the notion of restricted Minkowski sum of A and B with respect to Θ ⊂ A × B: A + Θ B = {x + y, (x, y) ∈ Θ}, and show that an analogue of the Brunn-Minkowski inequality holds: Theorem 1. There exists a positive constant c such that for all ρ ∈]0, 1[, n ∈ N, for all A, B ⊂ R n and Θ ⊂ A × B such that: ρ ≤ |A| |B| 1 n ≤ ρ −1 and |Θ| |A|.|B| ≥ 1 − c min(ρ √ n, 1),