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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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   We show that for any even log-concave probability measure $\mathrm{\mu <\#comment/>}$on ${\mathbb{R}}^n$, any pair of symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $\mathrm{\mu <\#comment/>}\left(\right(1-<\#comment/>\mathrm{\lambda <\#comment/>})K+\mathrm{\lambda <\#comment/>}L{)}^{c_n}\ge <\#comment/>(1-<\#comment/>\mathrm{\lambda <\#comment/>}\left)\mathrm{\mu <\#comment/>}\right(K{)}^{c_n}+\mathrm{\lambda <\#comment/>}\mathrm{\mu <\#comment/>}(L{)}^{c_n},$where $c_n\ge <\#comment/>n^{-<\#comment/>4-<\#comment/>o\left(1\right)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. 

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