On a linear refinement of the Prékopa-Leindler inequality

Abstract. If f , g are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater than or equal to the p-mean of the integrals of f and g. In this paper we prove that under the sole assumption that f and g have a common projection onto a hyperplane, the Prékopa–Leindler inequality admits a linear reûnement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.