Banach's isometric subspace problem in dimension four

TL;DR

This paper proves that in four-dimensional space, if all three-dimensional sections of a convex body are linearly equivalent, then the body must be a centered ellipsoid, solving a special case of Banach's isometric subspace problem.

Contribution

It provides the first resolution of Banach's problem for the case where the ambient space is four-dimensional and the subspaces are three-dimensional, using differential geometry.

Findings

Convex bodies with all 3D sections linearly equivalent are ellipsoids.

The result affirms Banach's conjecture in the specific case n=3, dim V=4.

Differential geometric methods are effective in this topologically challenging case.

Abstract

We prove that if all intersections of a convex body $B\subset\mathbb R^4$ 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 $2 \le n< \dim V$, 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.