The Brunn-Minkowski Inequality for Log-concave Probability Distributions

Abstract: The Brunn-Minkowski inequality is a fundamental inequality in geometry. The inequality states that for every compact set $A,B \subset \mathbb{R}^n$, and every $\lambda \in [0,1]$, one has $$ Vol(A+B)^{1/n} \geq Vol(A)^{1/n} + Vol(B)^{1/n}, $$ where $Vol$ denotes Lebesgue measure, and $A+B = \{a+b : a \in A, b \in B\}$ is the Minkowski sum of $A$ and $B$. The Brunn-Minkowski inequality has many applications in geometry, probability and analysis. In particular, it yields the famous isoperimetric inequality in a few lines, as well as the log-Sobolev inequality for the Gaussian distribution. It also has deep connections with information theory. The class of log-concave probability distributions is rich and contains important distributions in probability and statistics, such as the Gaussian distribution, the Laplace distribution, uniform distributions on a convex set, and gamma distributions of some parameters. In this talk, we will see that the Brunn-Minkowski inequality holds true for log-concave distributions under symmetry assumptions. We will then discuss some applications to new isoperimetric-type inequalities.