In this note we give an estimate in terms of upper curvature bounds for the volume growth of globally minimal submanifolds in Riemannian manifolds, new isoperimetric inequalities for these submanifolds, an explicit formula of the least volumes of closed submanifolds in symmetric spaces. As a result, we prove that every Helgason's sphere in a compact irreducible simply connected symmetric space is a globally minimal submanifold.