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We extend Prekopa’s Theorem and the Brunn-Minkowski Theo- rem from convexity to F-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of “har- monic interpolation” that we view as a generalization of Minkowski-addition.
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Suppose $$f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$$ is convex where $$\kappa\ge 0, \sigma>0$$, and the argmin function $$\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$$ exists and is single valued. We will prove $$\gamma$$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.more » « less
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null (Ed.)Suppose $$f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$$ is convex where $$\kappa\ge 0, \sigma>0$$, and the argmin function $$\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$$ exists and is single valued. We will prove $$\gamma$$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.more » « less
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