Uniqueness in the near isotropic Lp dual Minkowski problem

TL;DR

This paper proves the uniqueness of solutions to the near isotropic qth Lp dual Minkowski problem on the sphere for certain parameter ranges, establishing optimal estimates and extending previous results.

Contribution

It introduces new uniqueness results for the near isotropic Lp dual Minkowski problem, including optimal C0 estimates and conditions for solution uniqueness.

Findings

Proved uniqueness of solutions near isotropic conditions for -1<p<1 and q close to n.

Established optimal C0 estimates for -1<p<1.

Proved uniqueness for even solutions when -1<p<q<min{n,n+p} and q>0.

Abstract

For n>1 and -1<p<1, we prove that if q is close to n and the qth Lp dual curvature is Holder close to be the constant one function, then this "near isotropic" qth Lp dual Minkowski problem on the (n-1)-dimensional sphere has a unique solution. Along the way, we establish a C0 estimate for -1<p<1 that is optimal in the sense that if p<-1 and q=n, then it is known that the analogous C0 estimate does not hold. We also prove the uniqueness of the solution of the near isotropic even qth Lp dual Minkowski problem on the (n-1)-dimensional sphere if -1<p<q<min{n,n+p} and q>0.