On the number of solutions to the discrete two-dimensional \(L_{0}\)-Minkowski problem.
type: Item
instance of: scholarly article
title: On the number of solutions to the discrete two-dimensional \(L_{0}\)-Minkowski problem.
zbMATH Open document ID: 1054.52001
DOI: 10.1016/S0001-8708(03)00005-7
author: Alina Stancu
published in: Advances in Mathematics
publication date: 2004-01-06T00:00:00Z
review text: 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.
reviewed by: C. Peri
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 2022092
zbMATH Keywords: convex polygons
zbMATH Keywords: crystalline flows
zbMATH Keywords: Minkowski problem
MaRDI profile type: Publication
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full work available at URL: s0001-8708(03)00005-7
OpenAlex ID: W2079495504
description: scientific article; zbMATH DE number 2022092