  The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the
best-known open problems in the theory of posets, stating that the balance
constant (a quantity determined by the linear extensions) of any non-total
order is at least $1/3$. By reinterpreting balance constants of posets in terms
of convex subsets of the symmetric group, we extend the study of balance
constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture
that the lower bound of $1/3$ still applies in any finite Weyl group, with new
and interesting equality cases appearing.
  We generalize several of the main results towards the $1/3$-$2/3$ Conjecture
to this new setting: we prove our conjecture when $C$ is a weak order interval
below a fully commutative element in any acyclic Coxeter group (an
generalization of the case of width-two posets), we give a uniform lower bound
for balance constants in all finite Weyl groups using a new generalization of
order polytopes to this context, and we introduce generalized semiorders for
which we resolve the conjecture.
  We hope this new perspective may shed light on the proper level of generality
in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods
are likely to be successful in resolving it.
