Coxeter groups through the looking glass: Mirrors and Convexity

The convex hull operator on the configuration of mirrors of a Coxeter group is generated by repeated use of the convex hull operator restricted to collinear sets of mirrors (biconvex sets are line-biconvex sets). Each element of the possibly infinite group corresponds to a finite biconvex subset of the set M of mirrors. By appealing to the notion of line-biconvexity, we avail ourselves of a radical simplification of the combinatorial structure of Coxeter groups. A combinatorial formulation of the group law, the algorithmic construction of reduced words for group elements, and the lattice operations in the weak order, all become algorithmically feasible.