On the Carathéodory number for strong convexity

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

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