On the complexity of the set of unconditional convex bodies

TL;DR

This paper investigates the complexity of unconditional convex bodies in high-dimensional spaces, revealing exponential lower bounds on the size of separated sets and limitations on approximation by polytopes.

Contribution

It establishes exponential bounds on the size of separated sets of unconditional convex bodies and demonstrates the difficulty of approximating such bodies with polytopes.

Findings

Existence of exponentially large separated sets of unconditional convex bodies.

Limitations on approximating unconditional bodies with polytopes with fewer faces.

Upper bounds on the size of separated sets of symmetric 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 $\exp \exp (c(t) \log^2 n)$.