For given integers $n>2$ and $k\in \{2,3,\dots,n-1\}$ the question of Banach asks whether any $n$-dimensional real Banach space with isometric $k$-dimensional sections is a Hilbert space. For $k=2$ this is proved by Mazur. For given $k$ and infinite-dimensional space this follows from Dvoretzky almost spherical section theorem. Gromov (Izvestiya 1967 31(5)) proves this for any even $k$ and also for odd $k$ and $n\geqslant k+2$ (in complex case, for even $k$ and also for odd $k$ and $n\geqslant 2k+2$). So, in real case even-dimensional spaces with isometric hyperplanes remain (as far as I know) open. Gromov's proof uses algebraic topology of Grassmanians.

For given integers $n>2$ and $k\in \{2,3,\dots,n-1\}$ the question of Banach asks whether any $n$-dimensional real Banach space with isometric $k$-dimensional sections is a Hilbert space. For $k=2$ this is proved by H. Auerbach, S. Mazur, S. Ulam (here I am not completely sure: Gromov attributes this result to Mazur without direct reference). For given $k$ and infinite-dimensional space this follows from Dvoretzky almost spherical section theorem. Gromov (Izvestiya 1967 31(5)) proves this for any even $k$ and also for odd $k$ and $n\geqslant k+2$ (in complex case, for even $k$ and also for odd $k$ and $n\geqslant 2k+2$). So, in real case even-dimensional spaces with isometric hyperplanes remain (as far as I know) open. Gromov's proof uses algebraic topology of Grassmanians.