Uniqueness when the Lₚ curvature is close to be a constant for p [0, 1)

Abstract For fixed positive integer n, p 0, 1) p ∈ [ 0, 1), a (0, 1) a ∈ (0, 1), we prove that if a function g: {S}^n-1 {R} g: S n - 1 → R is sufficiently close to 1, in the Cᵃ C a sense, then there exists a unique convex body K whose Lₚ L p curvature function equals g. This was previously established for n=3 n = 3, p=0 p = 0 by Chen et al. (Adv Math 411 (A): 108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368: 107166, 2020). Related, we show that if p=0 p = 0 and n=4 n = 4 or n 3 n ≤ 3 and p [0, 1) p ∈ [ 0, 1), and the Lₚ L p curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies ^-1 g λ - 1 ≤ g ≤ λ, for some >1 λ > 1, then ₗ {ₒ^n-1}hK (x) C (p, ) max x ∈ S n - 1 h K (x) ≤ C (p, λ), for some constant C (p, ) >0 C (p, λ) > 0 that depends only on p and λ. This also extends a result from Chen et al. [10. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the Lₚ L p surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the Lₚ L p -Minkowksi problem, for -n - n p 0.