Motzkin decomposition of closed convex sets

M. A. Goberna, E. González, J. E. Martínez-Legaz, M. I. Todorov

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)


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. © 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)209-221
JournalJournal of Mathematical Analysis and Applications
Publication statusPublished - 1 Apr 2010


  • Closed convex sets
  • Linear inequality systems
  • Semi-infinite optimization


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