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

Abstract

For fixed positive integer n, \(p\in [0,1)\), \(a\in (0,1)\), we prove that if a function \(g:{\mathbb {S}}^{n-1}\rightarrow {\mathbb {R}}\) is sufficiently close to 1, in the \(C^a\) sense, then there exists a unique convex body K whose \(L_p\) curvature function equals g. This was previously established for \(n=3\), \(p=0\) 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 \(p=0\) and \(n=4\) or \(n\le 3\) and \(p\in [0,1)\), and the \(L_p\) curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies \(\lambda ^{-1}\le g\le \lambda \), for some \(\lambda >1\), then \(\max _{x\in {\mathbb {S}}^{n-1}}h_K(x)\le C(p,\lambda )\), for some constant \(C(p,\lambda )>0\) that depends only on p and \(\lambda \). 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 \(L_p\) surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the \(L_p\)-Minkowksi problem, for \(-n<p<0\).