  The Banach isometric conjecture asserts that a normed space with all of its
$k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first
case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an
elegant topological argument. We refine their method to arrive at a stable
version of their result: if all $2$-dimensional subspaces are almost isometric,
then the space is almost Euclidean. Furthermore, we show that a $2$-dimensional
surface, which is not a torus or a Klein bottle, equipped with a
near-monochromatic Finsler metric, is approximately Riemannian. The stability
is quantified explicitly using the Banach-Mazur distance.
