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

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.