On the number of solutions to the discrete two-dimensional \(L_{0}\)-Minkowski problem.
The discrete two-dimensional \(L_{o}\)-Minkowski problem asks for the existence and the uniqueness of a convex polygon of given outer normals to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices. This problem is an example of the so-called \(L_{p}\)-Minkowski problems which arise naturally in Lutwak's extension of the Brunn-Minkowski theory to the Brunn-Minkowski-Firey theory. In previous papers [Indiana Univ. Math. J. 45, 1157--1174 (1996; Zbl 0873.35034); Adv. Math. 167, No. 1, 160--174 (2002; Zbl 1005.52002)] the author showed the existence of a \(L_{o}\)-solution and, for centrally symmetric data, the uniqueness of the solution. In the present paper, she proves the uniqueness without the centrally symmetric assumption, the latter being an essential assumption in the uniqueness results for other \(L_{p}\)-Minkowski problems discussed in the literature. The method exploits the equivalence between \(L_{o}\)-solution and the homothetic solutions to appropriate crystalline flows. As a consequence, the author also gives a positive answer to a conjecture proposed by J. Taylor for crystalline flows.
scientific article; zbMATH DE number 2022092