  We give an improvement of the Carath\'eodory theorem for strong convexity
(ball convexity) in $\mathbb R^n$, reducing the Carath\'eodory number to $n$ in
several cases; and show that the Carath\'eodory 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.
