Equality cases of the Alexandrov--Fenchel inequality are not in the polynomial hierarchy

TL;DR

This paper proves that determining equality cases of the Alexandrov--Fenchel inequality for convex polytopes is computationally hard, unlikely to be in the polynomial hierarchy, linking geometric problems with complexity theory.

Contribution

It establishes the first complexity-theoretic hardness result for describing equality conditions of the Alexandrov--Fenchel inequality, using poset theory and order polytopes.

Findings

Equality cases are not in the polynomial hierarchy unless it collapses.

First hardness result linking geometric inequalities with computational complexity.

Uses poset theory and Stanley's order polytopes in the proof.

Abstract

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.