On the complexity of the set of unconditional convex bodies

Abstract

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 $\exp \exp (c(t){\log }^{2}n)$.

19 pages

References in corpus (1)

• On the metric entropy of the Banach-Mazur compactum