I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is, its intersections with planes through the origin) are affine equivalent planar bodies. Then $K$ is an ellipsoid.


Equivalently, if we consider the norm in $\mathbb R^3$ defined by $K$, this fact can be restated as follows: If a 3-dimensional normed space (with a smooth strictly convex norm) is such that all 2-dimensional linear subspaces are isometric to one another, then the normed space is Euclidean.


My motivations and my proof are from differential geometry, but the fact itself looks like something that might be studied in convex geometry. I know there are many ellipsoid characterization theorems but I could not find this particular one.


So the question is: Is the above fact known, and what are appropriate references? If it is known, does it generalize to non-smooth bodies and to higher dimensions?



Convex body with affine-equivalent cross-sections
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](http://link.springer.com/article/10.1007%2FBF01733278) (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](http://www.mathnet.ru/links/af24c946930b8cffe80c73af47c459f9/im2577.pdf) (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.


