  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.
