Local Blaschke-Kakutani ellipsoid characterization and Banach's isometric subspaces problem

Abstract

We prove the following local version of Blaschke-Kakutani's characterization of ellipsoids: Let V be a finite-dimensional real vector space, B subset of V a convex body with 0 in its interior, and 2 <= k < dim Van integer. Suppose that the body B is contained in a cylinder based on the cross-section B boolean AND 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. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).