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, $B\subset V$ 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.