On the Lp Minkowski Problem for Polytopes

Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1. As observed by Schneider [21], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R, with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classical Minkowski problem. For polytopes the problem asks for the necessary and sufficient conditions on a set of unit vectors u1, . . . , un ∈ Sd−1 and a set of real numbers α1, . . . , αn > 0 that guarantee the existence of a polytope, P , in R with n facets whose outer unit normals are u1, . . . , un and such that the facet whose outer unit normal is ui has area (i.e., (d − 1)-dimensional volume) αi. This problem was completely solved by Minkowski himself (see Schneider [21] for reference): If the unit vectors do not lie on a closed hemisphere of Sd−1, then a solution exists if and only if n ∑