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On a geometric combination of functions related to Pr\'ekopa-Leindler inequality by Crasta, Graziano; Fragalà, Ilaria is a scholarly article available to read on EtoBox.

What is On a geometric combination of functions related to Pr\'ekopa-Leindler inequality about?

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

Author

Crasta, Graziano; Fragalà, Ilaria

Published

2022

Language

EN