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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.16484 |
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Table of Contents:
- 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 $γ> 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_γ:= 1 - \frac{n+1}γ$. This result is sharp as $γ\searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $γ< \infty$. In particular, whenever $γ\leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold.