Uniqueness when the \(L_p\) curvature is close to be a constant for \(p\in[0, 1)\). (English) Zbl 07873545
Summary: 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 S. Chen et al. [Adv. Math. 411, Part A, Article ID 108782, 18 p. (2022; Zbl 1509.52006)] and in the symmetric case by S. Chen et al. [Adv. Math. 368, Article ID 107166, 20 p. (2020; Zbl 1440.52013)]. Related, we show that if \(p = 0\) and \(n = 4\) or \(n \leq 3\) and \(p\in[0, 1)\), and the \(L_p\) curvature function \(g\) of a (sufficiently regular, containing the origin) convex body \(K\) satisfies \(\lambda^{-1} \leq g \leq \lambda\), for some \(\lambda > 1\), then \(\max_{x\in\mathbb{S}^{n-1}}h_K(x) \leq C(p, \lambda)\), for some constant \(C(p, \lambda) > 0\) that depends only on \(p\) and \(\lambda\). This also extends a result from C. Chen and L. Xu [Adv. Math. 411, Part A, Article ID 108794, 27 p. (2022; Zbl 1505.35167)]. 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 \(-n < p < 0\).
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 52A39 | Mixed volumes and related topics in convex geometry |
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