On the uniqueness of solutions to the isotropic $L_p$ dual 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 $(p,q)\in (-n-1,-1]\times [n,n+1)$.

arXiv comment10 pages

References in corpus (8)

• The Logarithmic Minkowski Problem
• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• The dual Minkowski problem for negative indices
• The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
• The $L_p$-Minkowski problem for $-n<p<1$
• A unified flow approach to smooth, even $L_p$-Minkowski problems
• Uniqueness of solutions to a class of isotropic curvature problems
• Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem