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Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued C^1,1-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 methoddoi:10.1137/130945983 fatcat:mlrgpm6sn5euxd5sawmjbi5bdi