Motzkin decomposition of closed convex sets

Abstract

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. This paper provides five characterizations of the larger class of closed convex sets 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 in the paper. 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.

Motzkin predecomposable sets

“…1.3], it is easy to prove that G is contained in some maximal closed face e G. By assumption, the linear mapping x 7 ! Ax attains a lexicographical minimum over e G at some point c 2 C. On the other hand, by Proposition 14 and Corollary 9, the face e G is M-decomposable and, in view of the proof of [4,Proposition 16], the set convC \ af f e G is a compact component of e G. Since G is a face of e G (see [12, p. 163]), by Proposition 14 the set convC \ af f G = convC \ af f e G \ af f G is a compact component of G. Hence, by [4,Proposition 13. (v)], if a 2 R n is such that the linear function x 7 !…”

“…The following proposition will be useful for proving another property of Mpredecomposable sets mentioned in Section 1, namely that the result on the existence of a minimal decomposition of an M-decomposable set [4,Theorem 19] holds also for M-predecomposable sets.…”

“…More precisely, those sets in R n which can be written as the Minkowski sum of a compact and convex set and a closed and convex cone. Such sets were introduced in [4], where they were called Motzkin decomposable, or M-decomposable, in short, and further studied in [6] and [5], together with the M-decomposable functions, namely those whose epigraphs are M-decomposable. The class of M-decomposable sets lies hence in between the classes of closed polyhedra and of closed and convex sets.…”

On the Stability of the Motzkin Representation of Closed Convex Sets

“…The classical Motzkin Theorem [17] asserts that any polyhedral convex set is M-decomposable. This class of closed convex sets has been characterized in di¤erent ways in [7], [8] and [9]. For instance, a closed convex set F R n is M-decomposable i¤ F \ (lin F ) ?…”

“…its orthogonal complement). In that case, [7,Theorem 19] shows that F = cl conv M (F ) + 0 + F = cl M (F ) + 0 + F;…”