Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in ${\mathbb{R}}^n$ to have Carath'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath'{e}odory theorem and give a Tverberg type theorem for families of convex compacta.