Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist?
An equivalent statement:
Is it possible to find four $0$-symmetric convex bodies in the plane such that a three-dimensional $0$-symmetric convex body containing four central sections linearly equivalent to these four bodies does not exist?
This seems to be an old problem, but I'm not sure of its origin. The earliest reference I've found is
Rolewicz, S., Plane sections of centrally symmetric convex bodies. Israel J. Math. 4 1966 135–138.
Does anyone know the status of this problem? Some earlier references?
Just for completeness, let me add that Bessaga proved (C. Bessaga,A note on universal Banach spaces of a finite dimension, Bull. Acad. Polon. Sci.6 (1958), 97–101.) that
Given $n > 2$, it is possible to find $N = N(n)$ norms $\| \cdot\|_k$ $( 1 \leq k \leq N)$ on the plane such that an n-dimensional normed space containing $N$ subspaces isometric to these normed planes does not exist.
Rolewicz mentions that it is not clear whether we can take $N(n) = n+1$.
Added on 08/01/2014. Trying to guess what possible four normed planes cannot be linearly and isometrically embedded in any one three-dimensional normed space: Does there exist a three-dimensional centrally symmetric convex body that contains four central sections that are linearly equivalent to a circle, a square, a regular hexagon, and a regular octagon?