The Discrete Planar L0-Minkowski Problem

Abstract

In the discrete setting, the L 0 -Minkowski problem extends the question posed and answered by the classical Minkowski's existence theorem for polytopes. In particular, the planar extension, which we address in this paper, concerns the existence of a convex polygonal body which contains the origin, whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area.

The logarithmic Minkowski problem

“…In the plane (n = 2), the L p -Minkowski problem was treated by Stancu [57,58], Umanskiy [61], Chen [7], and by Jiang [32]. Solutions to the L p -Minkowski problem are the homothetic solutions of Gauss curvature flows (see e.g., [2], [3], [9], [11], [15]).…”

The Brunn-Minkowski inequality

“…Lutwak, Yang, and Zhang [107] study the L p version of the Minkowski problem (see Section 6). Stancu [140] treats a version of the L p -Minkowski problem corresponding to p = 0, related to an earlier investigation of Firey [59] …”

On the Lp Minkowski Problem for Polytopes

“…During the past decade various elements of the L p Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9] [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [24], [25], [26], [27], [28], [29]). …”

“…A normalized version (discussed below) of the problem was proposed and completely solved for p > 1 and even data in [19]. For d = 2, the important case p = 0 of the discrete-data L p Minkowski problem was dealt with by Stancu [26], [27].…”