  We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Amp\`ere
masses of plurisubharmonic functions, which generalizes a result of Demailly
and Pham. As applications to convex geometry, this gives a complex analytic
proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which
generalizes recent results in convex geometry of Kaveh-Khovanskii,
Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented)
Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric
plurisubharmonic functions in the Cegrell class, we confirm Demailly's
conjecture on the convergence of higher Lelong numbers under the canonical
approximation.
