Generalizations of the Lyndelöf theorem

The Lyndelöf theorem states: a polihedrom described near the ball has the least volume among all convex polyhedra of equal volume and of the same face direction. Generalizations of the Lyndelöf theorem made by A. D. Alexandrov, G. Hadviger and the author of this paper are presented. The Lyndelöf theorem generalizations made by A. D. Alexandrov and G. Hadviger are proved be equivalent. A generalization of the Lyndelöf theorem in the $n$-dimentional Minkovsky's $M^n$ ($n>2$) surface is formulated and proved.