Uniqueness When the Lp Curvature is Close to be a Constant for p ∈ [0, 1)

Böröczky, Károly J. and Saroglou, Christos (2024) Uniqueness When the Lp Curvature is Close to be a Constant for p ∈ [0, 1). CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. pp. 1-27. ISSN 0944-2669 (print); 1432-0835 (online) (In Press)

Preview

Text 2308.03367v3.pdf Download (326kB) | Preview

Abstract

For fixed positive integer n, p ∈ [0, 1), a ∈ (0, 1), we prove that if a function g : Sn−1 → R is sufficiently close to 1, in the Ca sense, then there exists a unique convex body K whose Lp curvature function equals g. This was previously established for n = 3, p = 0 by Chen, Feng, Liu and in the symmetric case by Chen, Huang, Li, Liu. Related, we show that if p = 0 and n = 4 or n ≤ 3 and p ∈ [0, 1), and the Lp curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies λ−1 ≤ g ≤ λ, for some λ > 1, then maxx∈Sn−1 hK(x) ≤ C(p, λ), for some constant C(p, λ) > 0 that depends only on p and λ. This also extends a result from Chen, Feng, Liu [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the Lp surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the Lp-Minkowksi problem, for −n<p<0.

Actions (login required)

Edit Item