$L^p$-Minkowski Problem under Curvature Pinching
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916813250494464 |
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| author | Ivaki, Mohammad N. Milman, Emanuel |
| author_facet | Ivaki, Mohammad N. Milman, Emanuel |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16484 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L^p$-Minkowski Problem under Curvature Pinching Ivaki, Mohammad N. Milman, Emanuel Differential Geometry Functional Analysis 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. |
| title | $L^p$-Minkowski Problem under Curvature Pinching |
| topic | Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2307.16484 |