  We reinterpret an inequality, due originally to Sidorenko, for linear
extensions of posets in terms of convex subsets of the symmetric group
$\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in
arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the
hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for
$B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map
closely related to the well-studied promotion operator on linear extensions;
this map may be of independent interest. We also note that the inequalities in
question can be interpreted as a triangle inequalities, so that convex hulls
can be used to define a new invariant metric on $W$ whenever our conjecture
holds. Geometric properties of this metric are an interesting direction for
future research.
