On a geometric combination of functions related to Prékopa–Leindler inequality

We introduce a new operation between nonnegative integrable functions on R n $\mathbb {R}^n$ , that we call geometric combination; it is obtained via mass transportation approach, playing with inverse distribution functions. The main feature of this the Lebesgue integral combination equals mean two separate integrals; as natural consequence, derive functional inequality Prékopa–Leindler type. When applied to characteristic measurable sets, their provides set whose volume volumes. In framework convex bodies, by comparing 0-sum, get an alternative proof log-Brunn–Minkowski for unconditional bodies and symmetries.