Uniqueness when the $L_p$ curvature is close to be a constant for $p\in[0,1)$

Abstract

For fixed positive integer $n$, $p\in[0,1]$, $a\in(0,1)$, we prove that if a function $g:\mathbb{S}^{n-1}\to \mathbb{R}$ is sufficiently close to 1, in the $C^a$ sense, then there exists a unique convex body $K$ whose $L_p$ curvature function equals $g$. This was previously established for $n=3$, $p=0$ by Chen, Feng, Liu \cite{CFL22} and in the symmetric case by Chen, Huang, Li, Liu \cite{CHLL20}. Related, we show that if $p=0$ and $n=4$ or $n\leq 3$ and $p\in[0,1)$, and the $L_p$ curvature function $g$ of a (sufficiently regular, containing the origin) convex body $K$ satisfies $\lambda^{-1}\leq g\leq \lambda$, for some $\lambda>1$, then $\max_{x\in\mathbb{S}^{n-1}}h_K(x)\leq C(p,\lambda)$, for some constant $C(p,\lambda)>0$ that depends only on $p$ and $\lambda$. This also extends a result from Chen, Feng, Liu \cite{CFL22}. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the $L_p$ surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $L_p$-Minkowksi problem, for $-n<p<0$.