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：描述亚历山德罗夫-芬切尔不等式的等式条件数十年来一直是一个重要的开放问题。本文证明在凸多面体情形下，除非多项式层级坍缩至有限层级，否则该问题的描述不属于多项式层级。这是该问题的首个硬度结果，也是对Shenfeld与van Handel近期研究成果（arXiv:archive/201104059）的复杂性理论对应——该研究给出了等式条件的几何特征描述。证明过程运用了斯坦利序多面体理论，并采用了偏序集理论工具。