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.
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