On the uniqueness of Lp-Minkowski problems: The constant p-curvature case in R^3

We study the C4 smooth convex bodies K ⊂Rn+1 satisfying K(x) =u(x)1−p, where x ∈Sn, K is the Gauss curvature of ∂K, u is the support function of K, and p is a constant. In the case of n =2, either when p ∈[−1, 0] or when p ∈(0, 1) in addition to a pinching condition, we show that K must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the Lp-Minkowski problem in R3. Moreover, we give an explicit pinching constant depending only on p when p ∈(0, 1).