Ivaki, M. N., & Milman, E. (2024). LP-Minkowski Problem Under Curvature Pinching. International Mathematics Research Notices, 2024(10), 8638–8652. https://doi.org/10.1093/imrn/rnad319
E104-06 - Forschungsbereich Konvexe und Diskrete Geometrie
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Journal:
International Mathematics Research Notices
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ISSN:
1073-7928
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Date (published):
May-2024
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Number of Pages:
15
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Publisher:
OXFORD UNIV PRESS
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Peer reviewed:
Yes
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Keywords:
Uniqueness in Lp-Minkowski problem; log-Minkowski inequality; curvature pinching; centro-affine geometry
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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 $\frac{1}{2}D^2 \|\|_{\ell K}^2$, 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.
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Project title:
Existenz und Eindeutigkeit von Lösungen für Krümmungsprobleme: P 36545-N (FWF - Österr. Wissenschaftsfonds)