Michael Roysdon
University of Cincinnati
Title
Generalized Supremal Convolutions
Abstract
The Brunn-Minkowski Inequality, which asserts that the Lebesgue measure on R^n is (1/n)-concave on the Borel sigma algebra on R^n with respect to the Minkowski Sum. It has been shown to have consequences in various areas of mathematics including convex geometry, geometric analysis, probability, and the geometry of Banach spaces.
The Prekopa-Leindler inequality (and more generally the Borell-Brascamp-Lieb inequality) is a functional variant of the Brunn-Minkowski inequality, acting as a reverse Holder type inequality, that has substantial applications dating back to the classical work of Prekopa, Leindler, and Brascamp-Lieb. One of the key features of the Prekopa-Leindler inequality is that it implies log-concave of the Guassian measure on R^n.
Often the Brunn-Minkowski inequality is applied to sets which have a very nice geometry structure, say convex, or symmetric, or both. In 2010 Gardner and Zvavitch ask if the concavity of the Gaussian measure can be improved to (1/n)-concavity under some restriction on the geometry of the sets and showed that this is true under certain restrictions. They also provided a 1-dimensional Borell-Brascamp-Lieb inequality , an improve of the Prekopa-Leindler inequality for the Gaussian measure under some restrictions, while leaving the general case open. It was left open as to whether this inequality may hold in a general dimension.
In this talk, we will discuss the Brunn-Minkowski inequality, its functional variants, and investigate the term "generalized supremal convolution", a functional addition vastly generalizing the classical Minkowski sum of sets. We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we affirmatively answer the question left by Gardner and Zvavitch, the existence of a Borell-Brascamp-Lieb inequality for the Gaussian measure and show that is equivalent to the geometric inequality it implies due to Eskenazis-Moschidis.
Based on a joint work with A. Malliaris, J. Melbourne, and C. Roberto