On the Caratheodory number for strong convexity

We give an improvement of the Caratheodory theorem for strong convexity (ball convexity) in $mathbb R^n$, reducing the Caratheodory number to $n$ in several cases; and show that the Caratheodory number cannot be smaller than $n$ for an arbitrary gauge body $K$. We also give an improved topological criterion for one convex body to be a Minkowski summand of another.