On the uniqueness of LpLp-Minkowski problems: The constant p -curvature case in R3R3

We study the C4C4 smooth convex bodies K⊂Rn+1K⊂Rn+1 satisfying K(x)=u(x)1−pK(x)=u(x)1−p, where x∈Snx∈Sn, K   is the Gauss curvature of ∂K∂K, u   is the support function of KK, and p   is a constant. In the case of n=2n=2, either when p∈[−1,0]p∈[−1,0] or when p∈(0,1)p∈(0,1) in addition to a pinching condition, we show that KK must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the LpLp-Minkowski problem in R3R3. Moreover, we give an explicit pinching constant depending only on p   when p∈(0,1)p∈(0,1).