Some functional inverse Santalo inequalities
Résumé
Let f : R-n --> R+ be a log-concave function and for Z is an element of R-n, define, f(z)(y) = inf(x is an element of Rn) e(-[x-z,y-z])/f(x) for every y is an element of R-n We discuss the problem of finding a sharp lower bound to the product P(f) = inf(z is an element of Rn) (integral(Rn)f(x)dx integral(Rn) f(z)(y)dy) We prove that if n = 1, then P(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) >= 4(n). These inequalities are functional versions, with exact lower bounds, of the so-called inverse Santalo inequality for convex bodies, that we state and discuss as conjectures. (C) 2008 Elsevier Inc. All rights reserved.