We show that 1. for every $A\subseteq \{0, 1\}^n$, there exists a polytope $P\subseteq \mathbb{R}^n$ with $P \cap \{0, 1\}^n = A$ and extension complexity $O(2^{n/2})$, 2. there exists an $A\subseteq \{0, 1\}^n$ such that the extension complexity of any $P$ with $P\cap \{0, 1\}^n = A$ must be at least $2^{\frac{n}{3}(1-o(1))}$. We also remark that the extension complexity of any 0/1-polytope in $\mathbb{R}^n$ is at most $O(2^n/n)$ and pose the problem whether the upper bound can be improved to $O(2^{cn})$, for $c<1$.

中文翻译：

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关于分离布尔立方体子集的多面体的扩展复杂性

我们表明1.对于每个$ A \ subseteq \ {0，1 \} ^ n $，存在一个多面体$ P \ subseteq \ mathbb {R} ^ n $与$ P \ cap \ {0，1 \} ^ n = A $和扩展复杂度$ O（2 ^ {n / 2}）$，2.存在一个$ A \ subseteq \ {0，1 \} ^ n $，这样任何$ P $的扩展复杂度$ P \ cap \ {0，1 \} ^ n = A $必须至少为$ 2 ^ {\ frac {n} {3}（1-o（1））} $。我们还注意到$ \ mathbb {R} ^ n $中任何0/1多义词的扩展复杂度最多为$ O（2 ^ n / n）$，并提出了上限是否可以提高到$的问题。 O（2 ^ {cn}）$，$ c <1 $。