L^p-Minkowski Problem under Curvature Pinching

2023/07/31 by Mohammad N. Ivaki, Emanuel Milman, Ivaki, Mohammad N. +1 · 2 citations
Mathematics · #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2307.16484

Abstract

Let K be a smooth, origin-symmetric, strictly convex body in ℝⁿ. If for some ℓ∈ GL(n,ℝ), the anisotropic Riemannian metric (1)/(2)D² \Vert⋅\Vert_{ℓ K}², encapsulating the curvature of ℓ K, is comparable to the standard Euclidean metric of ℝⁿ up-to a factor of γ> 1, we show that K satisfies the even L^p-Minkowski inequality and uniqueness in the even L^p-Minkowski problem for all p ≥ p_γ:= 1 - \fracn+1γ. This result is sharp as γ\searrow 1 (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all γ< ∞. In particular, whenever γ≤ n+1, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.

Cited by

• On p-Brunn-Minkowski and Brascamp-Lieb inequalities
• Concavity principles for weighted marginals

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