Quasiconvex polytope
A polyhedron is quasiconvex if all of the edges of its convex hull are also edges of P .^{[1]} Quasiconvexity is itself a broad term, but it is most interesting together with the constraints of regular faces, nonselfintersection, and no coplanar adjacent faces. It is suspected that the set of quasiconvex polyhedra under these conditions is finite (apart from polyhedra whose convex hulls are prisms or antiprisms).
Although quasiconvex polyhedra are not necessarily toroidal, quasiconvexity is primarily examined with respect to Stewart toroids. The concept of quasiconvexity was originally suggested by Norman Johnson as an attempt to restrict regular faced toroidal polyhedra to a finite set; without quasiconvexity, blending a Stewart toroid with any appropriate acrohedron produces another Stewart toroid, causing an explosion in possibilities that renders the set much less interesting. The property first appears as the property "(Q)" in Adventures Among the Toroids, written by Bonnie Stewart.
Related notions[edit  edit source]
Weak quasiconvexity[edit  edit source]
A polyhedron, P , is weakly quasiconvex if all of the edges of its convex hull are entirely coincident on the 1skeleton of P . This allows for edges in the convex edges which are coextant with multiple edges in P . Every quasiconvex polyhedron is weakly quasiconvex, but not every weakly quasiconvex polyhedron is quasiconvex. This property is called (Q') in Adventures Amoung the Toroids.

A polyhedron made of eight cubes that is (R) and (Q'), but not (Q) or (A). Some of the edges of its convex hull are made of 3 of its original edges.

A polyhedron that is (R)(A) and (Q') but not (Q). Some of the edges of its convex hull are made of 3 of its original edges.
Regularfaced quasiconvexity[edit  edit source]
A polyhedron, P , is regularfaced quasiconvex if it is quasiconvex and every face in the convex hull is regular. This property is called (Q") in Adventures Amoung the Toroids.^{[2]} Stewart conjectured that the number of polyhedra which were (R)(A)(Q)(T), but not (Q") is small and restricted his search only to polyhedra with the (Q") property. Others since have investigated polyhedra which are (R)(A)(Q)(T) but not (Q").

A polyhedron that is (R)(A) and has a regularfaced convex hull, but is not (Q).

A polyhedron from Adventures that is (R)(A) and (Q") (as well as (Q')), but not (Q), plus a view of its tunnel.
External links[edit  edit source]
 Doskey, Alex. "B.M. Stewart's  Adventures Among the Toroids  Index".
References[edit  edit source]
 ↑ Stewart (1964:77)
 ↑ Stewart (1964:79)
Bibliography[edit  edit source]
 Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686119 363.