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

We prove that if f : R n → [ 0 , ∞ ) is an integrable log-concave function with f ( 0 ) = 1 and F 1 , … , F r are linear subspaces of R n such that s I n = ∑ i = 1 r c i P i where I n is the identity operator and P i is the orthogonal projection onto F i , then n n ∫ R n f ( y ) n d y ⩾ ∏ i = 1 r ∫ F i f ( x i ) d x i c i / s . As an application we obtain the dual version of the Bollobás–Thomason inequality: if K is a convex body in R n with 0 ∈ int ( K ) and ( σ 1 , … , σ r ) is an s -uniform cover of [ n ], then | K | s ⩾ 1 ( n ! ) s ∏ i = 1 r | σ i | ! ∏ i = 1 r | K ∩ F i | . This is a sharp generalization of Meyer’s dual Loomis–Whitney inequality.