标题： 关于无条件凸体集的复杂性 显示英文标题

标题： On the complexity of the set of unconditional convex bodies

摘要： 我们证明对于任何$t>1$，在$\mathbb{R}^n$中的无条件凸体集合包含一个至少具有$\exp \exp (C(t) n)$基数的$t$-分离子集。 这表明在$\mathbb{R}^n$中存在一个不可约的凸体，除非$N > \exp(c(d)n)$，否则不能通过具有$N$个面的多面体的投影在距离$d$内进行近似。 我们还证明了对于$t>2$，完全对称体在$\mathbb{R}^n$中的$t$-分离集的基数不超过$\exp \exp (c(t) \log^2 n)$。 显示英文摘要

摘要： 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 $\exp \exp (c(t) \log^2 n)$.