  We prove two colorful Carath\'eodory theorems for strongly convex hulls,
generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre
B\'ar\'any, the non-colorful Carath\'eodory theorem for strongly convex hulls
by the second author, and the "very colorful theorems" by the first author and
others. We also investigate if the assumption of a "generating convex set" is
really needed in such results and try to give a topological criterion for one
convex body to be a Minkowski summand of another.
