On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski problem

TL;DR

This paper proves the uniqueness of smooth, strictly convex solutions to a specific isotropic $L_p$ dual Minkowski problem within a certain parameter range, establishing the unit sphere as the only solution.

Contribution

It establishes the uniqueness of solutions to the isotropic $L_p$ dual Minkowski problem for a defined parameter range, confirming the sphere as the sole smooth, strictly convex solution.

Findings

The unit sphere is the only smooth, strictly convex solution.

Uniqueness holds for $(p,q) ext{ in } (-n-1,-1] imes [n,n+1)$.

The result advances understanding of the geometric properties of the Minkowski problem.

Abstract

We prove that the unit sphere is the only smooth, strictly convex solution to the isotropic $L_p$ dual Minkowski problem \begin{align*} h^{p-1} |D h|^{n+1-q}\mathcal{K}=1, \end{align*} provided $(p,q)\in (-n-1,-1]\times [n,n+1)$.