Generous Sets

We investigate the notion of generosity, a particular case of non-selfishness. Let $\cal F$ be a family of sets in $\mathbb{R}^k$. A set $M \subset \mathbb{R}^k$ is called $\cal F$-convex if for any points $x,y\in M$ there is a set $F\in \cal F$ such that $x,y\in F$ and $F\subset M$. We call a family $\cal F$ of compact sets complete if $\cal F$ contains all compact $\cal F$-convex sets. A single convex body $K$ will be called generous, if the family of all convex bodies isometric to $K$ is not complete. We investigate here the generosity of convex bodies.