The Santalo point of a function, and a functional form of the Santalo inequality

Let L(f) denote the Legendre transform of a function f : R-n -> R. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f >= 0, there exists a translation J(x) = f (x - a) such that integral(Rn) e(-f) integral(Rn) e(-L(f)) <= (2 pi)(n). If a is selected so as to minimize the left side of (1), then equality in (1) is satisfied if and only if e(-f) is proportional to the distribution of a Gaussian random variable. This inequality immediately implies the Santalo inequality for convex bodies, as well as a new concentration inequality for the Gaussian measure.