Necessary and sufficient conditions for convexity and strong convexity, respectively, of connected sublevel sets that are defined by finitely many real-valued C-1,C-1-maps are presented. A novel characterization of strongly convex sets in terms of the so-called local quadratic support is proved. The results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous-time nonlinear systems to be strongly convex. An application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present.