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 centered ellipsoid. This gives an affirmative answer to the case $n=3$ following question by Banach from 1932: Is normed vector space $V$ whose $n$ -dimensional isometric, for fixed $2 \le n< \dim V$ , necessarily Euclidean? The dimensions and $\dim V=4$ first where was unresolved. Since 3-sphere parallelizable, known global topological methods do not help in this case. Our proof employs differential geometric approach.