Motzkin decomposition of closed convex sets via truncation

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

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11 Citations (Scopus)

Abstract

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 F̂ 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 F̂ 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 F̂. 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. © 2012 Elsevier Ltd.
Original languageEnglish
Pages (from-to)35-47
JournalJournal of Mathematical Analysis and Applications
Volume400
Issue number1
DOIs
Publication statusPublished - 1 Apr 2013

Keywords

  • Closed convex sets
  • Convex functions
  • Motzkin decomposition

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