On the number of solutions to the planar dual Minkowski problem

TL;DR

This paper investigates the dual Minkowski problem in the plane, determining the number of solutions for constant dual curvature when 0<q≤4, and improving nonuniqueness results for q>4, with applications to the Lp-Alexandrov problem.

Contribution

It provides a combined theoretical and numerical analysis to determine solution counts for the dual Minkowski problem in 2D, including new nonuniqueness results for q>4.

Findings

Number of solutions for 0<q≤4 determined

Improved nonuniqueness results for q>4

Applications to Lp-Alexandrov problem for p<0

Abstract

The dual Minkowski problem in the two-dimensional plane is studied in this paper. By combining the theoretical analysis and numerical estimation of an integral with parameters, we find the number of solutions to this problem for the constant dual curvature case when $0<q\leq4$. An improved nonuniqueness result when $q>4$ is also obtained. As an application, a result on the uniqueness and nonuniqueness of solutions to the $L_p$-Alexandrov problem is obtained for $p<0$.