On the complexity of the set of unconditional convex bodies

We show that for any $t>1$, the set of unconditional convex bodies in $\mathbb{R}^n$ contains a $t$-separated subset of cardinality at least $\exp \exp (C(t) n)$. This implies that there exists an unconditional convex body in $\mathbb{R}^n$ which cannot be approximated within the distance $d$ by a projection of a polytope with $N$ faces unless $N > \exp(c(d)n)$. We also show that for $t>2$, the cardinality of a $t$-separated set of completely symmetric bodies in $\mathbb{R}^n$ does not exceed $\exp \exp (c(t) \log^2 n)$.