On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in $\mathbb{R}^3$

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ in addition to a pinching condition, we show that $\mathbb{K}$ must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the $L_p$-Minkowski problem in $\mathbb{R}^3$. Moreover, we give an explicit pinching constant depending only on $p$ when $p\in(0,1)$.