The generalization of Minkowski problems for polytopes

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.