# Convex body with affine-equivalent cross-sections

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?

• So by "affine equivalent ", you mean up to a volume preserving linear transformation, i.e. an element of SL(n)? – Deane Yang Mar 11 '16 at 4:02
• For general dimensions it is a famous open problem, which was studied by young Gromov. For infinite dimensional Banach spaces if all infinite-dimensional subspaces are isomorphic, it is a Hilbert space indeed, proved by @gowers – Fedor Petrov Mar 11 '16 at 5:38
• @Deane: I mean that there are linear maps between 2-planes that send cross-sections one to another. The group of self-equivalences of a cross-section is indeed a subgroup of $SL(2,\mathbb R)$. – Sergei Ivanov Mar 11 '16 at 12:38

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.
• Yes, as well as for the cylinder over any planar convex set. I was thinking about the strictly convex case. I can prove the local version for smooth strictly convex bodies and $k=2$. – Sergei Ivanov Mar 11 '16 at 13:11