Banach's isometric subspace problem in dimension four

Abstract

We prove that if all intersections of a convex body B subset of R-4 with 3-dimensional linear subspaces are linearly equivalent then B is a centered ellipsoid. This gives an affirmative answer to the case n = 3 of the following question by Banach from 1932: Is a normed vector space V whose n-dimensional linear subspaces are all isometric, for a fixed 2 <= n < dim V, necessarily Euclidean? The dimensions n = 3 and dim V = 4 is the first case where the question was unresolved. Since the 3-sphere is parallelizable, known global topological methods do not help in this case. Our proof employs a differential geometric approach.