On the binary solitaire cone

The solitaire cone SB is the cone of all feasible fractional Solitaire Peg games. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. The link with the well studied dual metric cone and the similarities between their combinatorial structures see (3) leads to the study of a dual cut cone analogue; that is, the cone generated by the {0, 1}-valued facets of the solitaire cone. This cone is called binary solitaire cone and denoted BSB. We give some results and conjectures on the combinatorial and geometric properties of the binary solitaire cone. In particular we prove that the extreme rays of SB are extreme rays of BSB strengthening the analogy with the dual metric cone whose extreme rays are extreme rays of the dual cut cone. Other related cones are also considered.