On the Lp Minkowski Problem for Polytopes

Abstract

Abstract. Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L p Minkowski problem for all p > 1.As observed by Schneider [23], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R d , 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 u 1 , . . . , u n ∈ S d−1 and a set of real numbers α 1 , . . . , α n > 0 that guarantee the existence of a polytope, P , in R d with n facets whose outer unit normals are u 1 , . . . , u n and such that the facet whose outer unit normal is u i has area (i.e.,

The Lp Minkowski problem for polytopes for p<0

“…For the case where p > 1 with p = n, necessary and sufficient conditions for the existence of solutions to the discrete L p Minkowski problem were established by Hug, et al [30]. In Section 5, we give a new proof of this condition.…”

“…Establishing existence and uniqueness for the solution of the classical Minkowski problem was done by Aleksandrov, and Fenchel and Jessen (see, e.g., [52]). When p = 1, the L p Minkowski problem has been studied by, e.g., Lutwak [38], Lutwak and Oliker [39], Guan and Lin [18], Chou and Wang [10], Hug, et al [30], Böröczky, et al [5]. Additional references regarding the L p Minkowski problem and Minkowski-type problems can be found in [5, 8, 10, 17-21, 28-30, 32-34, 38, 39, 44, 53, 54].…”

“…One reason that the Minkowski and the L p Minkowski problem for polytopes are important is because the Minkowski problem and the L p Minkowski problem (for p > 1) for measures can be solved by an approximation argument by first solving the polytopal case (see, e.g., Hug, et al [30] and Schneider [52], pp. 392-393).…”

On the uniqueness ofLp-Minkowski problems: The constant p-curvature case inR3

“…Chou-Wang [12] solved (1.2) for a general measure when p > 1. Different proofs were presented in Hug-Lutwak-Yang-Zhang [24] for p > 1. C ∞ solution was given by Lutwak-Oliker [32] for the even case for p > 1.…”

Necessary subspace concentration conditions for the even dual Minkowski problem

“…It is also the subject of the central logarithmic Minkowski problem which asks for sufficient and necessary conditions of a measure µ on S n−1 to be the conevolume measure V K (·) of a convex body K ∈ K n o . This is the p = 0 limit case of the general L p -Minkowski problem within the above mentioned L p Brunn-Minkowski theory for which we refer to [32,38,57] and the references within.…”