  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 method to over-approximate
attainable sets when strong convexity is present.
