*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?

isometricsay to the Euclidean norm $\|\cdot\|$ and to $10000\|\cdot\|$ ? $\endgroup$ – Pietro Majer Aug 15 '16 at 16:08