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.
# Finite dimensional subspaces of $L^1.$
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 $f_1, \dots, f_d.$