  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.
