Advertisement

Compact Convex Sets with Prescribed Facial Dimensions

  • Vera Roshchina
  • Tian Sang
  • David Yost
Chapter
Part of the MATRIX Book Series book series (MXBS, volume 1)

Abstract

While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The ideas in this paper were motivated by the discussions that took place during a recent MATRIX program in approximation and optimisation held in July 2016. We are grateful to the MATRIX team for the enjoyable and productive research stay. We would also like to thank the two referees for their insightful corrections and remarks.

References

  1. 1.
    Alfsen, E.M., Shultz, F.W.: State Spaces of Operator Algebras: Basic Theory, Orientations, and C -products. Mathematics: Theory and Applications. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  2. 2.
    Eckhardt, U.: Theorems on the dimension of convex sets. Linear Algebra Appl. 12(1), 63–76 (1975)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Grünbaum, B.: The dimension of intersections of convex sets. Pac. J. Math. 12, 197–202 (1962)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hill, R.D., Waters, S.R.: On the cone of positive semidefinite matrices. Linear Algebra Appl. 90, 81–88 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Abridged version of convex analysis and minimization algorithms I and II. In: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin (2001).Google Scholar
  6. 6.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  7. 7.
    Pataki, G.: On the connection of facially exposed and nice cones. J. Math. Anal. Appl. 400(1), 211–221 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)Google Scholar
  9. 9.
    Sang, T.: Limit roots for some infinite Coxeter groups. Master’s Thesis, Department of Mathematics and Statistics, University of Melbourne (2014)Google Scholar
  10. 10.
    Tunçel, L.: Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization. Fields Institute Monographs, vol. 27. American Mathematical Society/Fields Institute for Research in Mathematical Sciences, Providence/Toronto (2010)Google Scholar
  11. 11.
    Vass, J.: On the Exact Convex Hull of IFS Fractals. arXiv:1502.03788v2 (2016)Google Scholar
  12. 12.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceRMIT UniversityMelbourneAustralia
  2. 2.Centre for Informatics and Applied OptimisationFederation University AustraliaBallaratAustralia
  3. 3.School of ScienceRMIT UniversityMelbourneAustralia

Personalised recommendations