Selfishness of convex bodies and discrete point sets

Let F be a family of sets in R d . A set M ⊂ R d 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 family F of compact sets is called complete if F contains all compact F -convex sets. Generalizing the definition in Yuan and Zamfirescu (2016), a compact set K will be called selfish, if the family F K of all sets similar to K contains all compact F K -convex sets. In this paper, we investigate the selfishness of rectangles, isosceles triangles, regular n -gons, and some finite sets.