= Choquet's theorem by strict convexity
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Maximize the integral of a continuous strictly convex <function> among <probability measures> with a fixed <barycenter>. The <supporting measure lemma for affine upper envelopes> implies that the maximizing <measure> has zero integral of the nonnegative envelope gap. Strict convexity makes this gap positive at every nonextreme point, so the <measure> is concentrated on the extreme boundary. Metrizability makes that boundary Borel and supplies the continuous strictly convex <function>.
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