L p -Minkowski Problem Under Curvature Pinching

Abstract

Abstract Let $K$ be a smooth, origin-symmetric, strictly convex body in ${\mathbb{R}}^{n}$. If for some $\ell \in \textrm{GL}(n,{\mathbb{R}})$, the anisotropic Riemannian metric $\frac{1}{2}D^{2} \left \Vert \cdot \right \Vert_{\ell K}^{2}$, 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.