This question is motivated by my discussion (via comments) with @fedja regarding [this earlier question.](https://mathoverflow.net/questions/94026/functions-with-same-area) In any case the question is whether there is any concise characterization of finite dimensional subspaces of $L^1.$ I found some papers by our own Bill Johnson on finite dimensional subspaces of $L^p$ spaces, but they appear to (a) be for $p>1$ and (b) use a lot of language hard to understand for us troglodytes. In the meantime, the question is very concrete: I give you a centrally symmetric convex body in $\mathbb{R}^d,$ and I ask whether this is the unit ball of the sup norm of the linear combinations of some $d$ functions $f\_1, \dots, f\_d.$


**EDIT** As @Fedor points out, I asked a different question from what I intended: I had meant to ask about $L^\infty,$ but the answers to this are very interesting, so I will let it stand, and I guess will ask a different question to avoid (or increase) confusion.



Finite dimensional subspaces of $L^1.$
I think that $K$ is such a convex body if and only if its dual $K^\*$ is a zonotope, i.e., a Minkowski sum of finitely many line segments, or a limit of zonotopes. I think that every limit $Z$ of zonotopes is characterized by a finite-mass measure $\mu$ on projective space $\mathbb{R}P^{d-1}$. Then the points in $Z$ are of the form
$$\int\_{S^{d-1}} \vec{x} d\sigma(\vec{x}),$$
which were $\sigma$ is a measure on the sphere $S^{d-1}$ that projects to $\mu$.


You can get every zonotope as an example of $K^\*$ by letting each $f\_k$ be piecewise constant. Then $K^\*$ is a projection of a finite-dimensional cube, and those are exactly the zonotopes. Then, every choice for $K^\*$ is a limit of zonotopes, because $f\_k$ has piecewise constant approximations. Actually you can make the measure $\mu$ directly by taking the pushforward of Lebesgue measure under the function $\vec{f}/||\vec{f}||$; that is a measure on the sphere but you can take its pushforward to projective space. A similar construction yields the measures $\sigma$.


