Convex-like Submanifolds, Caps and a Volume Comparison Theorem

Convex-like submanifolds in the Lorentz-Minkowski spaces are introduced in this paper as a natural and broad generalization of convex surfaces in the Euclidean space. Associated with such a convex-like submanifold in a Minkowski space, we define roofs (which resembles the supporting planes), caps (which are generated by the light rays in the normal bundle), and towers (which revives the notion of interior for such high codimensional objects). As a touchstone of this theory, a volume comparison theorem is proved, which is a non-trivial generalization of the obvious fact that among two nested convex surfaces, the outer one has larger volume. Finally we provide two important applications of this theorem.