Numerical solution of the Minkowski problem

摘要

We present a numerical procedure for solving the Minkowski problem, i.e., determining the convex set corresponding to a given curvature function. The method is based on Minkowski's isoperimetric inequality concerning convex and compact sets in R3. The support function of the target set is approximated in finite function space, so the problem becomes one of constrained optimization in Rn, which in turn is solved by Newtonian (or other) iteration. We prove some properties of the optimization function and the constraining set and present some numerical examples.