Banach's isometric subspace problem in dimension four

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.