Valuations on the Space of Quasi-Concave Functions

  • Andrea Colesanti
  • Nico Lombardi
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)


We characterize the valuations on the space of quasi-concave functions on \(\mathbb{R}^{N}\), that are rigid motion invariant and continuous with respect to a suitable topology. Among them we also provide a specific description of those which are additionally monotone.


Compact Support Convex Body Radon Measure Rigid Motion Convex Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford University Press, Oxford, 2000)zbMATHGoogle Scholar
  2. 2.
    Y. Baryshnikov, R. Ghrist, M. Wright, Hadwiger’s theorem for definable functions. Adv. Math. 245, 573–586 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. Cavallina, Non-trivial translation-invariant valuations on L . Preprint, 2015 (arXiv:1505.00089)Google Scholar
  4. 4.
    L. Cavallina, A. Colesanti, Monotone valuations on the space of convex functions. Anal. Geom. Metr. Spaces 1 (3) (2015) (electronic version)Google Scholar
  5. 5.
    H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin-Göttingen-Heidelberg, 1957)CrossRefzbMATHGoogle Scholar
  6. 6.
    D. Klain, A short proof of Hadwiger’s characterization theorem. Mathematika 42, 329–339 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Klain, G. Rota, Introduction to Geometric Probability (Cambridge University Press, New York, 1997)zbMATHGoogle Scholar
  8. 8.
    H. Kone, Valuations on Orlicz spaces and L ϕ-star sets. Adv. Appl. Math. 52, 82–98 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Ludwig, Fisher information and matrix-valued valuations. Adv. Math. 226, 2700–2711 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Ludwig, Valuations on function spaces. Adv. Geom. 11, 745–756 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Ludwig, Valuations on Sobolev spaces. Am. J. Math. 134, 824–842 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Ludwig, Covariance matrices and valuations. Adv. Appl. Math. 51, 359–366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    D. Ma, Analysis of Sobolev Spaces in the Context of Convex Geometry and Investigations of the Busemann-Petty Problem. PhD thesis, Technische Universität, Vienna, 2015Google Scholar
  14. 14.
    D. Ma, Real-valued valuations on Sobolev spaces. Preprint, 2015 (arXiv: 1505.02004)Google Scholar
  15. 15.
    M. Ober, L p-Minkowski valuations on L q-spaces. J. Math. Anal. Appl. 414, 68–87 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, 2nd expanded edn. (Cambridge University Press, Cambridge, 2014)zbMATHGoogle Scholar
  17. 17.
    A. Tsang, Valuations on L p-spaces. Int. Math. Res. Not. 20, 3993–4023 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    A. Tsang, Minkowski valuations on L p-spaces. Trans. Am. Math. Soc. 364 (12), 6159–6186 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Wang, Affine Sobolev inequalities. PhD thesis, Technische Universität, Vienna, 2013Google Scholar
  20. 20.
    T. Wang, Semi-valuations on \(BV (\mathbb{R}^{n})\). Indiana Univ. Math. J. 63, 1447–1465 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Wright, Hadwiger integration on definable functions, PhD thesis, University of Pennsylvania, 2011Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”FirenzeItaly

Personalised recommendations