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

For fixed positive integer n , p∈ [0,1) p ∈ [ 0 , 1 ) , a∈ (0,1) a ∈ ( 0 , 1 ) , we prove that if a function g:Sⁿ⁻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 λ ⁻¹≤ g≤ λ λ - 1 ≤ g ≤ λ , for some λ >1 λ > 1 , then max _x∈ Sⁿ⁻¹hK(x)≤ C(p,λ ) max x ∈ S n - 1 h K ( x ) ≤ C ( p , λ ) , for some constant C(p,λ )>0 C ( p , λ ) > 0 that depends only on p and λ 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 $$-n<p<0$$ - n < p < 0 .