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We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to$F$-subharmonicity. We apply this to the interpolation problem of convexfunctions and convex sets introducing a new notion of "harmonic interpolation"that we view as a generalization of Minkowski-addition.

Interpolation, Prekopa and Brunn-Minkowski for $F$-subharmonicity