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不等式的等号条件数十年来一直是一个重大未解决问题。我们证明，在凸多面体情形下，除非多项式层次坍缩至有限层，否则该描述不属于多项式层次。这是该问题的首个困难性结果，也是Shenfeld与van Handel近期成果（arXiv:archive/201104059）的复杂度对应物——后者给出了等号条件的几何刻画。证明过程涉及Stanley序多面体并运用了偏序集理论技术。