Integral inequalities for $\alpha$-convolutions of $\alpha$-concave functions

Abstract

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $\alpha$-concave functions under $\alpha$-convolutions, an operation that arises naturally in the ``geometrization of probability'' program. In particular, we establish a Pl\"unnecke--Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $\alpha$-convolutions of $\alpha$-concave functions. We also prove a sharp Rogers--Shephard-type inequality and characterize its equality cases for $\alpha$-concave functions, bridging the log-concave case studied by Alonso-Guti\'errez, Gonz\'alez-Merino, Jim\'enez, and Villa and the quasi-concave case studied by Colesanti.