Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D . In that case, the sets C and D are called compact and conic components of F . This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of F with closed halfspaces), when F contains no lines, and truncations of the intersection F ̂ of F with the orthogonal complement of the lineality of F , otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F ̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H . Thus, any Motzkin decomposable set F can be expressed as F = C + D , where the compact component C is a truncation of F ̂ . These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F . The minimality of this type of decompositions is also discussed.