Abstract

We prove that every convex cone V of a real vector space X possessing an uncountable Hamel basis may be expressed as the cone of all the half-lines contained within some convex subset C of X (in other words, V is the infinity cone to C). This property does not hold for lower-dimensional vector spaces; more precisely, a convex cone V in a vector space X with a denumerable basis is the infinity cone to some convex subset of X if and only if V is the union of a countable ascending sequence of linearly closed cones, while a convex cone V in a finite-dimensional vector space X is the infinity cone to some convex subset of X if and only if V is linearly closed.

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Keywords

  • Infinity cone
  • recession analysis
  • spreading cover

Mathematics Subject Classification

26B99, 46N10, 49J99

References

  1. H. Attouch, G. Buttazzo, G. Michaille: Variational Analysis in Sobolev and BV Spaces, Series on Optimization 6, SIAM, Philadelphia (2006).Google Scholar
  2. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer, New York (2002).Google Scholar
  3. J. Borwein, R. Goebel: Notions of relative interior in Banach spaces, J. Math. Sci., New York 115 (2003) 2542–2553.https://doi.org/10.1023/A:1022988116044Google Scholar
  4. N. Bourbaki: Eléménts de Mathématique XX, Hermann & Cie., Paris (1967).Google Scholar
  5. A. Daniilidis, J.-E. Martinez-Legaz: Characterizations of evenly convex sets and evenly quasiconvex functions, J. Math. Anal. Appl. 273 (2002) 58–66.https://doi.org/10.1016/S0022-247X(02)00206-8Google Scholar
  6. E. Ernst, M. Volle: Infinity cones and functions in real vector spaces: an existence result, Pac. J. Optim. 4 (2008) 465–481.Google Scholar
  7. F. Jongmans: Cris et chuchotements des cônes, Bull. Soc. Roy. Sci. Liège 49 (1980) 312–346.Google Scholar
  8. V. Klee: Convex sets in linear spaces, Duke Math. J. 18 (1951) 443–466.https://doi.org/10.1215/S0012-7094-51-01835-2Google Scholar
  9. V. Klee, E. Maluta, C. Zanco: Basic properties of evenly convex sets, J. Convex Analysis 14 (2007) 137–148.https://doi.org/10.68381/jca14011Google Scholar
  10. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  11. E. Steinitz: Bedingt konvergente Reihen und konvexe Systeme. (Teil I.), J. reine angew. Math. 143 (1913) 128–175.https://doi.org/10.1515/crll.1913.143.128Google Scholar

Suggested citation

E. Ernst, M. Volle. “When is a Convex Cone the Cone of all the Half-Lines Contained in a Convex Set?.” Journal of Convex Analysis 16 (2009), No. 3&4, 749–766. https://doi.org/10.68381/jca16045

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.