On a geometric combination of functions related to Pr\'ekopa-Leindler inequality

Graziano Crasta; Ilaria Fragalà

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arXiv · 2204.11521

On a geometric combination of functions related to Pr\'ekopa-Leindler inequality

Graziano Crasta·Ilaria Fragalà

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Abstract

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.

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BibTeX RIS

Graziano Crasta, Ilaria Fragalà. 2022-04-25. On a geometric combination of functions related to Pr\'ekopa-Leindler inequality. https://arxiv.org/abs/2204.11521

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