Sharp stability of functional and geometric inequalities

Seminar in Mathematics

Abstract: The Brunn-Minkowski inequality is a fundamental result in convex geometry and analysis, closely related to the isoperimetric inequality. It states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Here A+B={x+y : x \in A, y \in B}. Equality holds if and only if A and B are homothetic and convex sets in R^d.

The Prekopa-Leindler inequality is a functional generalization of the Brunn-Minkowski inequality with important applications to high dimensional probability theory. If t \in (0,1) and f,g,h : R^d -> R_+ are continuous functions with bounded support such that h(z) = \sup_{z = tx + (1-t)y} f^t(x) g^{1-t}(y),  then  \int h dx \geq (\int f dx)^t  (\int g dx)^{1-t}. Equality holds if and only if f and g are homothetic (i.e. f=ag(x+b)) and log-concave (i.e. \log(f) is concave). The Borell-Brascamp-Lieb inequality is a strengthening of the Prekopa-Leindler inequality, replacing the geometric mean with other means.

The stability of these inequalities has been intensely studied lately. The stability of the Brunn-Minkowski inequality states that if we are close to equality, then A and B must be close to being homothetic and convex. Similarly, the stability of the Prekopa-Leindler and Borell-Brascamp-Lieb inequalities states that if we are close to equality, then f and g must be close to being homothetic and concave. In this talk, we present sharp stability results for the Brunn-Minkowski, Prekopa-Leindler and Borell-Brascamp-Lieb inequalities, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on these problems.

This talk is based on joint work with Alessio Figalli and Peter van Hintum.