$L^p$-Minkowski Problem under Curvature Pinching

TL;DR

This paper establishes sharp conditions under which convex bodies satisfy the even $L^p$-Minkowski inequality and uniqueness, based on curvature pinching conditions related to an anisotropic metric, extending classical results.

Contribution

It introduces a curvature pinching condition that guarantees the even $L^p$-Minkowski inequality and uniqueness for a range of p, improving classical inequalities and characterizing ellipsoids in the limit.

Findings

Validates the inequality for all p ≥ p_γ

Shows sharpness as γ approaches 1

Extends results to the log-Minkowski case when γ ≤ n+1

Abstract

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$, is comparable to the standard Euclidean metric of $\mathbb{R}^{n}$ up-to a factor of $\gamma > 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_\gamma := 1 - \frac{n+1}{\gamma}$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma < \infty$. In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.