Motzkin decomposition of closed convex sets via truncation

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