Reverse Brascamp–Lieb inequality and the dual Bollobás–Thomason inequality

Abstract

We prove that if $f:{\mathbb {R}}^n\rightarrow [0,\infty )$ is an integrable log-concave function with $f(0)=1$ and $F_1,\ldots ,F_r$ are linear subspaces of ${\mathbb {R}}^n$ such that $sI_n=\sum _{i=1}^rc_iP_i$ where $I_n$ is the identity operator and $P_i$ is the orthogonal projection onto $F_i$, then

$$\begin{aligned} n^n\int \limits _{{\mathbb {R}}^n}f(y)^ndy\geqslant \prod _{i=1}^r\left( \,\int \limits _{F_i}f(x_i)dx_i\right) ^{c_i/s}. \end{aligned}$$

As an application we obtain the dual version of the Bollobás–Thomason inequality: if K is a convex body in ${\mathbb {R}}^n$ with $0\in \mathrm{int}(K)$ and $(\sigma _1,\ldots ,\sigma _r)$ is an s-uniform cover of [n], then

$$\begin{aligned} |K|^s\geqslant \frac{1}{(n!)^s}\prod _{i=1}^r|\sigma _i|!\prod _{i=1}^r|K\cap F_i|. \end{aligned}$$

This is a sharp generalization of Meyer’s dual Loomis–Whitney inequality.