A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.

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TheLpAleksandrovintegralcurvatureanditscorrespondingchar- acterization problem—the Lp Aleksandrov problem—were recently introduced by Huang, Lutwak, Yang, and Zhang. The current work presents a solution to the Lp Aleksandrov problem for origin-symmetric polytopes when −1 < p < 0.

The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy- Kubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.