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 also discuss some related problems to Condorcet domains.