Right Quadruple Convexity of Complements

Let F be a family of sets in Rd (always d≥2). A set M⊂Rd is called F-convex, if for any pair of distinct points x,y∈M, there is a set F∈F, such that x,y∈F and F⊂M. A set of four points {w,x,y,z}⊂Rd is called a rectangular quadruple, if conv{w,x,y,z} is a non-degenerate rectangle. If F is the family of all rectangular quadruples, then we obtain the right quadruple convexity, abbreviated as rq-convexity. In this paper we focus on the rq-convexity of complements, taken in most cases in balls or parallelepipeds.