On the cross-product conjecture for the number of linear extensions

We prove a weak version of the cross–product conjecture: $F(k+1,\mathrm{\ell })F(k,\mathrm{\ell }+1)\ge (\frac{1}{2}+\epsilon )F(k,\mathrm{\ell })F(k+1,\mathrm{\ell }+1)$, where $F(k,\mathrm{\ell })$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\mathrm{\ell }$ apart, respectively, and where $\epsilon >0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross–product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words.