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

By studying L^p-combinations of strongly isomorphic polytopes, we give a simple proof of the equivalence of the L^p-Brunn-Minkowski inequality conjectured by Böröczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman. In addition, we prove the local inequality in dimension 2 for p=0, yielding a new proof of the log-Brunn-Minkowski inequality in the plane.