  We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell)
{F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where
${F}(k,\ell)$ is the number of linear extensions for which the values at fixed
elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where
$\varepsilon>0$ 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.
