Quasi-convexity

Quasi-convexity is a property of polyhedra related to convexity. It is primarily examined with respect to Stewart toroids, however there are spherical polyhedra which are quasi-convex but not convex (e.g. excavated icosahedron).

Definition
A polyhedron, $P$, is quasi-convex if all of the edges of its convex hull are also edges of $P$.

History
The concept of quasi-convexity was originally suggested by Norman Johnson as a restriction on regular faced toroidal polyhedra to a finite set. The property first appears as the property "(Q)" in Adventures Amoung the Toroids, written by Bonnie Stewart.

Weak quasi-convexity
A polyhedron, $P$, is weakly quasi-convex if all of the edges of its convex hull are entirely coincident on the 1-skeleton of $P$. This allows for edges in the convex edges which are coextant with multiple edges in $P$. Every quasi-convex polyhedron is weakly quasi-convex, but not every weakly quasi-convex polyhedron is quasi-convex. This property is called (Q') in Adventures Amoung the Toroids.

Regular-faced quasi-convexity
A polyhedron, $P$, is regular-faced quasi-convex if it is quasi-convex and every face in the convex hull is regular. This property is called (Q") in Adventures Amoung the Toroids. 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").