Uniqueness when the $$L_p$$ curvature is close to be a constant for $$p\in [0,1)$$

Abstract

For fixed positive integer n, $$p\in [0,1)$$ p ∈ [ 0 , 1 ) , $$a\in (0,1)$$ a ∈ ( 0 , 1 ) , we prove that if a function $$g:{\mathbb {S}}^{n-1}\rightarrow {\mathbb {R}}$$ g : S n - 1 → R is sufficiently close to 1, in the $$C^a$$ C a sense, then there exists a unique convex body K whose $$L_p$$ 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 $$n\le 3$$ n ≤ 3 and $$p\in [0,1)$$ p ∈ [ 0 , 1 ) , and the $$L_p$$ L p curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies $$\lambda ^{-1}\le g\le \lambda $$ λ - 1 ≤ g ≤ λ , for some $$\lambda >1$$ λ > 1 , then $$\max _{x\in {\mathbb {S}}^{n-1}}h_K(x)\le C(p,\lambda )$$ max x ∈ S n - 1 h K ( x ) ≤ C ( p , λ ) , for some constant $$C(p,\lambda )>0$$ C ( p , λ ) > 0 that depends only on p and $$\lambda $$ 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 $$L_p$$ L p surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $$L_p$$ L p -Minkowksi problem, for $$-n<p<0$$ - n < p < 0 .