Motzkin decomposition of closed convex sets via truncation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:400

Article

Goberna, M. A.¹  Iusem, A.²  Martinez-Legaz, J. E.³  Todorov, M. I.⁴ 

[1] Univ Alicante, Dept Stat & Operat Res, Alicante, Spain

[2] Inst Matematica Pura & Aplicada, Rio De Janeiro, Brazil

[3] Univ Autonoma Barcelona, Dept Econ & Econ Hist, Barcelona, Spain

[4] Univ Amer, Dept Actuary & Math, Puebla, Mexico

关键词: Motzkin decomposition;    Closed convex sets;    Convex functions;   

DOI  :  10.1016/j.jmaa.2012.10.072

来源: Elsevier

【 摘 要 】

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 (F) over cap 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 (F) over cap 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 (F) over cap . 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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free