On the Complexity of the Set of Unconditional Convex Bodies

We show that for any 1 ≤ t ≤ c̃n log−5/2 n, the set of unconditional convex bodies in R contains a t-separated subset of cardinality at least exp ( exp ( c t2 log(1 + t) n )) . This implies the existence of an unconditional convex body in R 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 R does not exceed exp ( exp ( C log n log t )) .