  Having in mind the well known model of Euclidean convex hypersurfaces [4],
[5], and the ideas in [1] many authors defined and investigate convex
hypersurfaces of a Riemannian manifold. As it was proved by the first author in
[7], there follows the interdependence between convexity and Gauss curvature of
the hypersurface. In this paper we define $H$-convexity of a Riemannian
submanifold of arbitrary codimension, replacing the normal versor of a
hypersurface with the mean curvature vector $H$. A characterization, used by
B.Y. Chen [2], [3] as the definition of strictly $H$-convexity, it is obtained.
