A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled. In this paper we show that this is best possible: in every dimension there exist two nonempty polytopes such that if a formulation for the convex hull of their union has a number of inequalities that is polynomial in the number of inequalities that describe the two polytopes, then the number of additional variables is at least linear in the dimension of the polytopes. We then show that this result essentially carries over if one wants to approximate the convex hull of the union of two polytopes and also in the more restrictive setting of lift-and-project.

中文翻译：

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多胞体联合的 Balas 公式是最佳的

一个著名的巴拉斯定理给出了两个非空多胞体的并集的线性混合整数公式，其松弛给出了这个并集的凸包。Balas 公式中的不等式数量与描述两个多胞体的不等式数量呈线性关系，并且变量数量加倍。在本文中，我们证明了这是最好的可能：在每个维度中都存在两个非空多面体，如果它们并集的凸包的公式有许多不等式，这些不等式是描述两个多面体的不等式数量的多项式，那么附加变量的数量至少在多胞体的维度上是线性的。