TITLE: Finite dimensional subspaces of $L^1.$
QUESTION [5 upvotes]: This question is motivated by my discussion (via comments) with @fedja regarding this earlier question. 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.

REPLY [3 votes]: Greg is right, of course, the dual must be a zonotope.
Let me mention also a direct characterization: for any vectors $x_1,\dots,x_k,y_1,\dots,y_m$ such that $\sum |f(x_i)|\geq \sum |f(y_j)|$ for any linear functional $f$, one may deduce that $\sum \|x_i\|\geq \sum \|y_j\|$. For Euclidean norm it is usual stuff in integral geometry: for proving inequality between sums of lengths one proves corresponding inequality for sums of projections to the same line. The norms, for which such trick is possible, are exactly those coming from sections of $L^1$.