We introduce a new operation between nonnegative integrable functions on Rn$\mathbb {R}<^>n$, 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 Prekopa-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.
On a geometric combination of functions related to Prekopa-Leindler inequality
Graziano Crasta;Ilaria Fragala
2023-01-01
Abstract
We introduce a new operation between nonnegative integrable functions on Rn$\mathbb {R}<^>n$, 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 Prekopa-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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
