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On highly eccentric cones

  • Alberto SeegerEmail author
  • Mounir Torki
Original Paper

Abstract

This paper addresses the issue of estimating the largest possible eccentricity in the class of proper cones of \(\mathbb {R}^n\). The eccentricity of a proper cone is defined as the angle between the incenter and the circumcenter of the cone. This work establishes also various geometric and topological results concerning the concept of eccentricity.

Keywords

Convex cone Incenter Circumcenter Eccentricity of a proper cone 

Mathematics Subject Classification (2010)

52A20 52A40 

References

  1. Astorino, A., Gaudioso, M., Seeger, A.: An illumination problem: optimal apex and optimal orientation for a cone of light. J. Global Optim. (2013, online April). doi: 10.1007/s10898-013-0071-0
  2. Barker, G.P., Carlson, D.: Generalizations of top-heavy cones. Linear Multilinear Algebra 8, 219–230 (1979/1980)Google Scholar
  3. Barker, G.P., Foran, J.: Self-dual cones in Euclidean spaces. Linear Algebra Appl. 13, 147–155 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Edmonds, A.L., Hajja, M., Martini, H.: Coincidences of simplex centers and related facial structures. Beiträge Algebra Geom. 46, 491–512 (2005)MathSciNetzbMATHGoogle Scholar
  5. Fiedler, M., Haynsworth, E.: Cones which are topheavy with respect to a norm. Linear Multilinear Algebra 1, 203–211 (1973)MathSciNetCrossRefGoogle Scholar
  6. Goffin, J.-L.: The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5, 388–414 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Gourion, D., Seeger, A.: Solidity indices for convex cones. Positivity 16, 685–705 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Henrion, R., Seeger, A.: On properties of different notions of centers for convex cones. Set-Valued Var. Anal. 18, 205–231 (2010a)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Henrion, R., Seeger, A.: Inradius and circumradius of various convex cones arising in applications. Set-Valued Var. Anal. 18, 483–511 (2010b)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Henrion, R., Seeger, A.: Condition number and eccentricity of a closed convex cone. Math. Scand. 109, 285–308 (2011)MathSciNetzbMATHGoogle Scholar
  11. Iusem, A., Seeger, A.: Pointedness, connectedness, and convergence results in the space of closed convex cones. J. Convex Anal. 11, 267–284 (2004)MathSciNetzbMATHGoogle Scholar
  12. Iusem, A., Seeger, A.: Axiomatization of the index of pointedness for closed convex cones. Comput. Appl. Math. 24, 245–283 (2005a)MathSciNetzbMATHGoogle Scholar
  13. Iusem, A., Seeger, A.: On pairs of vectors achieving the maximal angle of a convex cone. Math. Program. 104(Ser. B), 501–523 (2005b)Google Scholar
  14. Iusem, A., Seeger, A.: Searching for critical angles in a convex cone. Math. Program. 120(Ser. B), 3–25 (2009)Google Scholar
  15. Kelly, L.M., Murty, K.G., Watson, L.T.: CP-rays in simplicial cones. Math. Program. 48(Ser. B), 387–414 (1990)Google Scholar
  16. López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)CrossRefzbMATHGoogle Scholar
  17. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  18. Seeger, A., Torki, M.: Centers of sets with symmetry or cyclicity properties. TOP, online August 2013. doi: 10.1007/s11750-013-0289-5
  19. Stern, R.J., Wolkowicz, H.: Invariant ellipsoidal cones. Linear Algebra Appl. 150, 81–106 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Walkup, D.W., Wets, R.J.B.: Continuity of some convex-cone-valued mappings. Proc. Am. Math. Soc. 18, 229–235 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2013

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’AvignonAvignonFrance
  2. 2.Université d’Avignon, CERIAvignonFrance

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