Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. This notion has been studied for several decades, motivated by its relevance for combinatorial optimisation problems. It is an important question to understand the extent to which the extension complexity of a polytope is controlled by its dimension, and in this paper we prove three different results along these lines. First, we prove that for a fixed dimension $d$, the extension complexity of a random $d$-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic $n$-vertex polygon (whose vertices lie on a circle) has extension complexity at most $24\sqrt n$. This bound is tight up to the constant factor $24$. Finally, we show that there exists an $n^{o(1)}$-dimensional polytope with at most $n$ facets and extension complexity $n^{1-o(1)}$.

中文翻译：

---

低维多胞体的扩展复杂度

有时，可以将复杂的多面体表示为更简单的多面体的投影。为了量化这种现象，polytope $P$ 的扩展复杂度被定义为（可能是更高维的）polytope 中的最小面数，从中可以获得 $P$ 作为（线性）投影。这个概念已经研究了几十年，其动机是它与组合优化问题的相关性。了解多胞体的扩展复杂性受其维度控制的程度是一个重要问题，在本文中，我们沿着这些方向证明了三种不同的结果。首先，我们证明对于固定维度 $d$，随机$d$维多胞体（作为球或球体上随机点的凸包获得）的扩展复杂度通常与其顶点数的平方根有关。其次，我们证明任何循环$n$-顶点多边形（其顶点位于圆上）的扩展复杂度最多为$24\sqrt n$。这个界限紧到常数因子 $24$。最后，我们证明存在一个 $n^{o(1)}$ 维多面体，最多具有 $n$ 个面和扩展复杂度 $n^{1-o(1)}$。