Local Blaschke--Kakutani ellipsoid characterization and Banach's isometric subspaces problem
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| Format: | Preprint |
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2023
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| _version_ | 1866913799842299904 |
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| author | Ivanov, Sergei Mamaev, Daniil Nordskova, Anya |
| author_facet | Ivanov, Sergei Mamaev, Daniil Nordskova, Anya |
| contents | We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ a convex body with 0 in its interior, and ${2\le k<\dim V}$ an integer. Suppose that the body $B$ is contained in a cylinder based on the cross-section $B \cap X$ for every $k$-plane $X$ from a connected open set of linear $k$-planes in $V$. Then in the region of $V$ swept by these $k$-planes $B$ coincides with either an ellipsoid, or a cylinder over an ellipsoid, or a cylinder over a $k$-dimensional base.
For $k=2$ and $k=3$ we obtain as a corollary a local solution to Banach's isometric subspaces problem: If all cross-sections of $B$ by $k$-planes from a connected open set are linearly equivalent, then the same conclusion as above holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12231 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local Blaschke--Kakutani ellipsoid characterization and Banach's isometric subspaces problem Ivanov, Sergei Mamaev, Daniil Nordskova, Anya Metric Geometry 46C15, 52A21 We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ a convex body with 0 in its interior, and ${2\le k<\dim V}$ an integer. Suppose that the body $B$ is contained in a cylinder based on the cross-section $B \cap X$ for every $k$-plane $X$ from a connected open set of linear $k$-planes in $V$. Then in the region of $V$ swept by these $k$-planes $B$ coincides with either an ellipsoid, or a cylinder over an ellipsoid, or a cylinder over a $k$-dimensional base. For $k=2$ and $k=3$ we obtain as a corollary a local solution to Banach's isometric subspaces problem: If all cross-sections of $B$ by $k$-planes from a connected open set are linearly equivalent, then the same conclusion as above holds. |
| title | Local Blaschke--Kakutani ellipsoid characterization and Banach's isometric subspaces problem |
| topic | Metric Geometry 46C15, 52A21 |
| url | https://arxiv.org/abs/2311.12231 |