The generalization of Minkowski problems for polytopes

Abstract

The generalization of Minkowski problems, such as the \(L_p\) and Orlicz Minkowski problems, have caused wide concern recently. In this paper, we will establish the existence of the Orlicz Minkowski problem for polytopes. In particular, a solution to the \(L_p\) Minkowski problem for polytopes with \(p>1\) is given. By the uniqueness of this solution, we present a new proof of the \(L_p\) Minkowski inequality that demonstrates the relationship between these two fundamental theorems of the \(L_p\) Brunn–Minkowski theory.