On Banach's isometric subspace problem

An old problem by Banach asks whether a finite-dimensional Banach space $V$ is necessarily Euclidean if all its hyperplanes are isometric to one another. An equivalent formulation is whether a centered convex body is necessarily an ellipsoid if all its central cross-sections are affine equivalent. The problem has been solved in some dimensions but the general case remains open. We will discuss a differential geometric approach to the problem, its connections to Finsler geometry, and a recent solution of the case $\dim V=4$, obtained jointly with D.Mamaev and A.Nordskova.