$L^p$-Minkowski Problem under Curvature Pinching

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 $p\ge p_{\gamma }:=1-\frac{n+1}{\gamma }$. 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 $\gamma \le n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.

arXiv comment19 pages. Final version, to appear in International Mathematics Research Notices

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Cited by in corpus (1)

• Stability of the cone-volume measure with near constant density