Theodore Motzkin proved, in 1936, that any polyhedral convex set can be
expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. We
have provided several characterizations of the larger class of closed convex sets,
Motzkin decomposable, in finite dimensional Euclidean spaces which are the sum
of a compact convex set with a closed convex cone. These characterizations
involve different types of representations of closed convex sets as the support
functions, dual cones and linear systems whose relationships are also analyzed. The
obtaining of information about a given closed convex set F and the parametric
linear optimization problem with feasible set F from each of its different
representations, including the Motzkin decomposition, is also discussed. Another
result establishes that a closed convex set is Motzkin decomposable if and only if
the set of extreme points of its intersection with the linear subspace orthogonal to
its lineality is bounded. We characterize the class of the extended functions whose
epigraphs are Motzkin decomposable sets showing, in particular, that these
functions attain their global minima when they are bounded from below. Calculus
of Motzkin decomposable sets and functions is provided.
