Local Blaschke--Kakutani ellipsoid characterization and Banach's isometric subspaces problem
Sergei Ivanov, Daniil Mamaev, Anya Nordskova
TL;DR
This work develops a local analogue of the Blaschke--Kakutani ellipsoid characterization within finite-dimensional real normed spaces and applies it to a local form of Banach's isometric subspaces problem. The authors introduce a local contracting/cylindricity framework, prove a local dichotomy in dimension three using projective-geometry arguments, and extend the result to higher dimensions by induction and dimension-reduction lemmas. When specialized to $k=2$ or $k=3$, the local characterization yields that cross-sections that are linearly equivalent must arise from either a Euclidean (ellipsoidal) structure or a cylindrical/cylindrically-degenerate base, thereby providing a local solution to Banach's problem. The approach blends convex-geometric analysis, quadratic-form reconstruction, and projective geometry, and interacts with recent work on local isometric subspaces to yield a non-symmetric generalization and practical local-dichotomy results.
Abstract
We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ 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.