The discrete planar \(L_0\)-Minkowski problem.(English)Zbl 1005.52002
The \(L_p\)-Minkowski problem, arising in Lutwak’s Brunn-Minkowski-Firey theory of \(p\)-sums of convex bodies as an analogue to the classical Minkowski problem, is treated here for \(p= 0\) and planar polygons. This problem concerns the existence of a convex polygon in \(\mathbb{R}^2\), containing the origin, for which the direction of each edge and the area of the triangle formed by the edge and the origin are prescribed. If no two edges are parallel, the solution always exists. For the case where parallel edges occur, sufficient conditions for the existence are given. It is also shown that centrally symmetric data lead to centrally symmetry polygons. The proofs employ crystalline flows. For questions of uniqueness, a separate paper is announced.
Reviewer: Rolf Schneider (Freiburg i.Br.)
MSC:
| 52A10 | Convex sets in \(2\) dimensions (including convex curves) |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
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