Let F be a family of sets in IRd. A set M⊂F 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. A convex body K is called selfish, if the family FK of all convex bodies similar to K is complete.
We say that a convex body C has the Rupert property if we can make a hole large enough in C to permit another copy of C to pass through.
In this talk we'll discuss the selfishness and Rupert property of some convex bodies.