The Hibi–Li (2016) conjecture is fully resolved

Abstract

Hibi and Li conjectured that the order polytope $\mathcal{O}(P)$ of a finite poset $P$ has at most as many faces of each dimension as its chain polytope $\mathcal{C}(P)$, and that equality in some dimension $k$ with $1\leq k\leq|P|-1$ forces the two polytopes to be unimodularly equivalent. Fourier proved the inequality in a 2026 preprint. Let $X$ be the five-element poset in which an element lies above two incomparable elements and below two other incomparable elements. Using two results from Fourier's preprint, we show that if an $n$-element poset contains $X$ as an induced subposet, then, for every $k$ with $2\leq k\leq n-1$, its chain polytope has at least $\binom{n-3}{k-2}$ more $k$-dimensional faces than its order polytope, and that for every $n\geq5$ some $n$-element poset containing $X$ attains all these bounds at once. Since Hibi and Li showed that $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent exactly when $P$ does not contain $X$, this proves the equality statement for these values of $k$. The second version of Fourier's preprint gives an independent proof of this statement. For $k=1$, the equality statement is false: the polytopes $\mathcal{O}(X)$ and $\mathcal{C}(X)$ have the same number of edges (Hibi, Li, Sahara and Shikama) but are not unimodularly equivalent. Together with Fourier's inequality, these results settle the conjecture. The proofs given in this paper are formally verified in Lean 4.