Pointedness, Connectedness, and Convergence Results in the Space of Closed Convex Cones

Alfredo Iusem; Alberto Seeger

doi:10.68381/jca11018

Abstract

Let

\mathcal{K}(H)

denote the collection of all closed convex cones in a finite dimensional real Hilbert space H. Painleve-Kuratowski convergence in this space has been the subject of numerous publications. However, the link between convergence and pointedness has not been explored in its full extent. This note contributes to close this gap. Some results concerning the connected components of

\mathcal{K}(H)

are also considered.

Author information

Contact details are reproduced from the original publication and may be historical.

Alfredo Iusem

Inst. de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro,
Brazil

iusp@impa.br

Alberto Seeger

University of Avignon, Dept. of Mathematics, 33 rue Louis Pasteur, 84000 Avignon, France

alberto.seeger@univ-avignon.fr

Keywords

pointed cone
solid cone
Painlevé-Kuratowski convergence
polarity
connectedness

Mathematics Subject Classification

52 A 07, 54 F 65

References

J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer Acad. Press, Dordrecht, Netherlands (1993).https://doi.org/10.1007/978-94-015-8149-3 Google Scholar
J. W. Gaddum: A theorem on convex cones with applications to linear inequalities, Proc. Amer. Math. Soc. 3 (1952) 957–960.https://doi.org/10.1090/S0002-9939-1952-0052129-8 Google Scholar
D. H. Jacobson: Extensions of Linear-Quadratic Control, Optimization and Matrix Theory, Academic Press, London (1977).Google Scholar
D. T. Luc, J. P. Penot: Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001) 4095–4121.https://doi.org/10.1090/S0002-9947-01-02664-2 Google Scholar
J. J. Moreau: Décomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires, C.R. Acad. Sci. Paris 255 (1962) 238–240.Google Scholar
R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar
A. Seeger: Degree of pointedness of a convex function, Bull. Austral. Math. Soc. 53 (1996) 159–167.https://doi.org/10.1017/S000497270001683X Google Scholar
A. Seeger, M. Torki: On eigenvalues induced by a cone-constraint, Linear Alg. and Appl. 372 (2003) 181–206.https://doi.org/10.1016/S0024-3795(03)00553-6 Google Scholar
D. W. Walkup, R. J.-B. Wets: Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967) 229–235.https://doi.org/10.1090/S0002-9939-1967-0209806-6 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.

[1] J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar

[2] G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer Acad. Press, Dordrecht, Netherlands (1993).https://doi.org/10.1007/978-94-015-8149-3 Google Scholar

[3] J. W. Gaddum: A theorem on convex cones with applications to linear inequalities, Proc. Amer. Math. Soc. 3 (1952) 957–960.https://doi.org/10.1090/S0002-9939-1952-0052129-8 Google Scholar

[4] D. H. Jacobson: Extensions of Linear-Quadratic Control, Optimization and Matrix Theory, Academic Press, London (1977).Google Scholar

[5] D. T. Luc, J. P. Penot: Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001) 4095–4121.https://doi.org/10.1090/S0002-9947-01-02664-2 Google Scholar

[6] J. J. Moreau: Décomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires, C.R. Acad. Sci. Paris 255 (1962) 238–240.Google Scholar

[7] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

[8] A. Seeger: Degree of pointedness of a convex function, Bull. Austral. Math. Soc. 53 (1996) 159–167.https://doi.org/10.1017/S000497270001683X Google Scholar

[9] A. Seeger, M. Torki: On eigenvalues induced by a cone-constraint, Linear Alg. and Appl. 372 (2003) 181–206.https://doi.org/10.1016/S0024-3795(03)00553-6 Google Scholar

[10] D. W. Walkup, R. J.-B. Wets: Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967) 229–235.https://doi.org/10.1090/S0002-9939-1967-0209806-6 Google Scholar