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 Bollobas–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.