Minimal point of a Gaussian measure

概要

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Let $ \mu $ be a first order Borel probability measure on a real locally convex Hausdorff space $ E $. A point m is called a minimal point of $ \mu $ if it minimizes the functional $ \phi_{\lambda}^{\mu} (z) = \int_E \| {z-x} \|_{\lambda} d {\mu}(x) $ for all measurable semi-norm $ \| \|_{\lambda} $ on $ E $. We prove that if $ \mu $ is a centered Gaussian $ \mu $-regular measure, then $ 0 $ is the unique minimal point of $ \mu $. On the other hand, let $ E $ be a dual Banach space, $ \| \|_E $ be the dual Banach norm and $ \mu $ be a Gaussian Radon measure for the $ w^{\ast} $-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 $ \mu $ is extensible to a Radon measure for the strong topology, then the $ s $-minimal point is uniquely the barycenter. But if $ \mu $ is not extensible, the $ s $-minimal point may not be unique. In the case where $ E = 1^{\infty} $, we give a sufficient condition for the uniqueness of the $ s $-minimal point and an example where the uniqueness does not hold.
九州大学の論文