Some functional inverse Santaló inequalities

Let f:Rn→R+ be a log-concave function and for z∈Rnz∈Rn, definefz(y)=infx∈Rne−〈x−z,y−z〉f(x)for every y∈Rn. We discuss the problem of finding a sharp lower bound to the productP(f)=infz∈Rn(∫Rnf(x)dx∫Rnfz(y)dy). We prove that if n=1n=1, then P(f)⩾eP(f)⩾e and characterize the case of equality. The same method allows to give a new simple proof of the fact that if f is sign-invariant, then for all n  , P(f)⩾n4P(f)⩾4n. These inequalities are functional versions, with exact lower bounds, of the so-called inverse Santaló inequality for convex bodies, that we state and discuss as conjectures.