Describing the equality conditions of the Alexandrov--Fenchel inequality has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem, and is a complexity counterpart of the recent result by Shenfeld and van Handel (arXiv:archive/201104059), which gave a geometric characterization of the equality conditions. The proof involves Stanley's order polytopes and employs poset theoretic technology.

翻译：描述Alexandrov–Fenchel不等式的等号成立条件数十年来一直是未解决的重要开放问题。我们证明，在凸多胞体情形下，除非多项式层级坍缩至有限层级，否则该描述不包含于多项式层级内。这是该问题首个困难性结果，也是Shenfeld与van Handel（arXiv:archive/201104059）近期工作的复杂度对应——后者给出了等号成立条件的几何刻画。证明过程涉及Stanley序多胞体，并运用了偏序集理论技术。