The case of equality for an inverse Santalo functional inequality
Résumé
Let f : R(n) -> R(+) be a log-concave function such that f(x(1), ... , x(n)) = f(vertical bar x(1)vertical bar, ... , vertical bar x(n)vertical bar) for every (x(1), ..., x(n)) is an element of R(n) and 0 < integral(Rn) f(x) dx < +infinity. It was proved in [4] and [5] that if for y is an element of R(n), f*(y) := inf(x is an element of Rn) e(-< x, y >)/f(x), then P(f) := integral(Rn) f(x)dx integral(Rn)f*(y)dy >= 4(n), We characterize here the case of equality: one has P(f) = 4(n) if and only if f can be written as f(x(1), x(2)) = e(-parallel to x1 parallel to K1)1(K2)(x(2)), where K(1) subset of R(n1) and K(2) subset of R(n2), n(1) + n(2) = n, are unconditional convex bodies such that P(K(i)) := vertical bar K(i)vertical bar vertical bar K(i)*vertical bar = 4(n)i/n(i)!, i = 1, 2, where K(i)* denotes the polar of K(i). These last bodies were characterized in [7] and [8].