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By studying $L^p$-combinations of strongly isomorphic polytopes, we prove theequivalence of the $L^p$-Brunn-Minkowski inequality conjectured byB\"or\"oczky, Lutwak, Yang and Zhang to the local version of the inequalitystudied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman,settling a conjecture of the latter authors. In addition, we prove the localinequality in dimension $2$, yielding a new proof of the $L^p$-Brunn-Minkowskiinequality in the plane.

Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality