MINIMAL POINT OF A GAUSSIAN MEASURE

Abstract

Let μ be a first order Borel probability measure on a real locally convex Hausdorff space E. A point m is called a minimal point of μ if it minimizes the functional $ \\phi_\\lambda^\\mu (z) = \\int_E \\| z-x \\|_E d\\mu(x) $for all measurable semi-norm $\\| \\|_\\lambda$ on E. We prove that if μ is a centered Gaussian μ-regular measure, then 0 is the unique minimal point of μ. On the other hand, let E be a dual Banach space, $\\| \\|_E$ be the dual Banach norm and μ be a Gaussian Radon measure for the w*-topology. Then m is called s-minimal if it minimizes the functional $ \\Phi_\\mu (z) = \\int_E \\| z-x \\|_E d\\mu(x). $ $We prove that if μ is extensible to a Radon measure for the strong topology, then the s-minimal point is uniquely the barycenter. But if μ is not extensible, the s-minimal point may not be unique. In the case where E= 1∞, we give a sufficient condition for the uniqueness of the s-minimal point and an example where the uniqueness does not hold.