Generous Sets

We investigate the notion of generosity, a particular case of non-selfishness. Let

be a family of sets in

\mathbb{R}^k R k

. A set

M \subset \mathbb{R}^k M ⊂ R k

is called

\cal F F

-convex if for any points

x,y\in M x , y ∈ M

there is a set

F\in \cal F F ∈ F

such that

x,y\in F x , y ∈ F

and

F\subset M F ⊂ M

. We call a family

\cal F F

of compact sets complete if

\cal F F

contains all compact

\cal F F

-convex sets. A single convex body

K K

will be called generous, if the family of all convex bodies isometric to

K K

is not complete. We investigate here the generosity of convex bodies.