On Rq-convexity

A set M C Rd is called rq-convex if, for any two distinct points x, y ∈ M, there are z, w ∈ M such that conv{x, y, z, w} is a non-degenerate rectangle.In this thesis we first discribe some non-discrete rq-convex sets according to the definition, and then probe into the rq-convexity about discrete point sets in the plane and the vertex sets of Platonic solids. The main results are as follows:(1) Discuss the rq-convexity of closed unbounded sectors in the plane, squares and the complement of any bounded set in Rd (d≥2); give a necessary and sufficient condition on the difference set of a triangle being rq-convex; and provide some examples of non-connected sets which are not rq-convex. (2) Study the rq-convexity of eleven Archimedean tilings (infinite sets), and prove that the vertex sets of Archimedean tilings (36), (44), (63), (3.6.3.6), (34.6), (32.4.3.4), (4.82) are rq-convex sets, while the vertex sets of Archimedean tilings (33.42), (3.4.6.4), (4.6.12), (3.122) are not. (3) Investigate the rq-convexity about finite point sets in the plane, and obtain an upper bound of the rq-convex completion number of any finite point set; characterize the configurations of 6-point rq-convex sets and 8-point rq-convex sets, and prove that the smallest odd cardinality of an rq-convex set in R2 is 9. (4) Investigate the rq-convexity about the vertex sets of Platonic solids in R3, and prove that the vertex set of the regular tetrahedron is not rq-convex and find its rq-convex completion number is 3.