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