Centers and partial volumes of convex cones I. Basic theory

  • Alberto SeegerEmail author
  • Mounir Torki
Original Paper


We study the concept of least partial volume of a proper cone in \(\mathbb {R}^n\). This notion is a reasonable alternative to the classical concept of solid angle. In tandem, we study the concept of volumetric center of a proper cone. We compare this kind of center with the old notion of incenter.


Partial volume of a convex cone Solid angle Volumetric center Incenter 

Mathematics Subject Classification

51M25 52A38 47L07 


  1. Barker, G.P., Carlson, D.: Generalizations of top-heavy cones. Linear Multilinear Algebra 8, 219–230 (1979/80)Google Scholar
  2. Bhattacharya, R., Fung, J., Murray, R.M., Tiwari, A.: Ellipsoidal cones and rendezvous of multiple agents. In: 43rd IEEE Conference on Decision and Control, Nassau (2004)Google Scholar
  3. Faraut, J., Korányi, A.: Analysis on symmetric cones. Oxford University Press, New York (1994)zbMATHGoogle Scholar
  4. Fiedler, M., Haynsworth, E.: Cones which are topheavy with respect to a norm. Linear Multilinear Algebra 1, 203–211 (1973)CrossRefMathSciNetGoogle Scholar
  5. Gardner, R.J.: Geometric tomography. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  6. Gourion, D., Seeger, A.: Deterministic and stochastic methods for computing the volumetric moduli of convex cones. Comput. Appl. Math. 29, 215–246 (2010)zbMATHMathSciNetGoogle Scholar
  7. Gourion, D., Seeger, A.: Solidity indices for convex cones. Positivity 16, 685–705 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21, 860–885 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  9. Henrion, R., Seeger, A.: On properties of different notions of centers for convex cones. Set Valued Var. Anal. 18, 205–231 (2010a)Google Scholar
  10. Henrion, R., Seeger, A.: Inradius and circumradius of various convex cones arising in applications. Set Valued Var. Anal. 18, 483–511 (2010b)Google Scholar
  11. Henrion, R., Seeger, A.: Condition number and eccentricity of a closed convex cone. Math. Scand. 109, 285–308 (2011)zbMATHMathSciNetGoogle Scholar
  12. Iusem, A., Seeger, A.: Normality and modulability indices. I. Convex cones in normed spaces. J. Math. Anal. Appl. 338, 365–391 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. Nesterov, Yu., Nemirovskii, A.S.: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  14. Ribando, J.M.: Measuring solid angles beyond dimension three. Discrete Comput. Geom. 36, 479–487 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  16. Seeger, A.: Epigraphical cones II. J. Convex Anal. 19, 1–21 (2012)zbMATHMathSciNetGoogle Scholar
  17. Seeger, A.: Aperture angle analysis for ellipsoids. Elec. Linear Algebra 26, 732–741 (2013)zbMATHMathSciNetGoogle Scholar
  18. Seeger, A.: Lipschitz and Hölder continuity results for some functions of cones. Positivity 18, 505–517 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  19. Seeger, A., Torki, M.: Centers of sets with symmetry or cyclicity properties. TOP 22, 716–738 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  20. Stern, R., Wolkowicz, H.: Invariant ellipsoidal cones. Linear Algebra Appl. 150, 81–106 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. Truong, V.A., Tuncel, L.: Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers. Math. Progr. Ser. A, 100, 295–316 (2004)Google Scholar
  22. Vinberg, E.B.: The theory of convex homogeneous cones. Trans. Moskow Math. 12, 340–403 (1963)zbMATHGoogle Scholar
  23. Walkup, D.W., Wets, R.J.-B.: Continuity of some convex-cone-valued mappings. Proc. Am. Math. Soc. 18, 229–235 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  24. Wills, M.D.: Hausdorff distance and convex sets. J. Convex Anal. 14, 109–117 (2007)zbMATHMathSciNetGoogle Scholar

Copyright information

© The Managing Editors 2014

Authors and Affiliations

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

Personalised recommendations