Inscribed and circumscribed ellipsoidal cones: volume ratio analysis

  • Alberto SeegerEmail author
  • Mounir Torki
Original Paper


Let K be a pointed closed convex cone with nonempty interior in \(\mathbb {R}^n\). The inscribed ellipsoidal cone in K, denoted by \(E^{\uparrow }(K)\), is defined as the unique ellipsoidal cone that maximizes the volume among all ellipsoidal cones contained in K. The cone \(E^{\uparrow }(K)\) can be viewed as an optimal inner approximation of K. One of the goals of this work is to analyze which percentage of the volume of K is captured by \(E^{\uparrow }(K)\). In particular, we show that the lowest possible percentage occurs when K is a simplicial cone. Our result corresponds to a conic version of a celebrated theorem of Ball (J Lond Math Soc 44:351–359, 1991) on the volume ratio function of a convex body. The problem of analyzing the quality of the outer approximation of K by means of its circumscribed ellipsoidal cone can be treated by using duality arguments.


Convex cone Inscribed ellipsoidal cone Circumscribed ellipsoidal cone Volume ratio Axial symmetry 

Mathematics Subject Classification

51M25 52A38 47L07 


  1. Ball, K.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. 44, 351–359 (1991)MathSciNetCrossRefGoogle Scholar
  2. Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry, pp. 1–58, Math. Sci. Res. Inst. Publ., vol. 31. Cambridge University Press, Cambridge (1997)Google Scholar
  3. Barthe, F.: On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134, 335–361 (1998)MathSciNetCrossRefGoogle Scholar
  4. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)zbMATHGoogle Scholar
  5. Gigena, S.: Integral invariants of convex cones. J. Differ. Geom. 13, 191–222 (1978)MathSciNetCrossRefGoogle Scholar
  6. Grünbaum, B.: Measures of symmetry for convex sets. 1963 Proc. Sympos. Pure Math., vol. VII, pp. 233–270. American Mathematical Society, ProvidenceGoogle Scholar
  7. Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21, 860–885 (1996)MathSciNetCrossRefGoogle Scholar
  8. Güler, O., Gürtuna, F.: Symmetry of convex sets and its applications to the extremal ellipsoids of convex bodies. Optim. Methods Softw. 27, 735–759 (2012)MathSciNetCrossRefGoogle Scholar
  9. Horwitz, A.: An area inequality for ellipses inscribed in quadrilaterals. J. Math. Inequal. 4, 431–443 (2010)MathSciNetCrossRefGoogle Scholar
  10. Jerónimo-Castro, J., Mcallister, T.B.: Two characterizations of ellipsoidal cones. J. Convex Anal. 20, 1181–1187 (2013)MathSciNetzbMATHGoogle Scholar
  11. Seeger, A., Torki, M.: Centers and partial volumes of convex cones. I. Basic theory. Beitr. Algebra Geom. 56, 227–248 (2015)MathSciNetCrossRefGoogle Scholar
  12. Seeger, A., Torki, M.: Centers and partial volumes of convex cones. II. Advanced topics. Beitr. Algebra Geom. 56, 491–514 (2015)MathSciNetCrossRefGoogle Scholar
  13. Seeger, A., Torki, M.: Conic version of Loewner–John ellipsoid theorem. Math. Program. 155, 403–433 (2016)MathSciNetCrossRefGoogle Scholar
  14. Seeger, A., Torki, M.: Measuring axial symmetry in convex cones. J. Convex Anal. 25, 3 (2018)MathSciNetzbMATHGoogle Scholar
  15. Seeger, A., Vidal-Nuñez, J.: Measuring centrality and dispersion in directional datasets: the elliposidal cone covering approach. J. Global Optim. 68, 279–306 (2017)MathSciNetCrossRefGoogle Scholar
  16. Truong, V.A., Tuncel, L.: Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers. Math. Program. 100(Ser. A), 295–316 (2004)MathSciNetCrossRefGoogle Scholar
  17. Vinberg, E.B.: The theory of convex homogeneous cones. Trans. Moskow Math. 12, 340–403 (1963)zbMATHGoogle Scholar

Copyright information

© The Managing Editors 2018

Authors and Affiliations

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

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