2023/11/20 by Sergei Ivanov, Ivanov, Sergei, Daniil Mamaev +3
Mathematics · Computer Science · #Point processes and geometric inequalities #Optimization and Variational Analysis #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2311.12231
We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let V be a finite-dimensional real vector space, B⊂ V a convex body with 0 in its interior, and 2≤ k<dim V an integer. Suppose that the body B is contained in a cylinder based on the cross-section B ∩ 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.