On Monotone Translation–Projection Covariant Minkowski Valuations

  • Yun Xu
  • Qi GuoEmail author


We study Minkowski valuations compatible with translations and projections. We first introduce the concept of translation–projection covariance for Minkowski valuations. Then, we show that, under some conditions, monotone translation–projection covariant Minkowski valuations are exactly orthogonal projections, which gives a characterization of the orthogonal projection operators on Euclidean spaces.


Minkowski valuation Orthogonal projection Translation Covariance Convex body 

Mathematics Subject Classification

52A20 52B45 



The authors express sincere thanks to the reviewers for their careful reading the first and the second versions of this paper, pointing out some language errors and for their valuable suggestions and comments which improved the paper.


  1. 1.
    Alesker, S.: Continuous rotation invariant valuations on convex sets. Ann. Math. 149(3), 977–1005 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alesker, S.: On P. McMullen’s conjecture on translation invariant valuations. Adv. Math. 155(2), 239–263 (2000)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Haberl, C.: Blaschke valuations. Am. J. Math. 33(3), 717–751 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Haberl, C., Parapatits, L.: Moments and valuations. Am. J. Math. 138(6), 1575–1603 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hadwiger, H.: Additive Funktionale $k$-dimensionaler Eikörper. I. Arch. Math. 3, 470–478 (1952)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hadwiger, H.: Additive Funktionale $k$-dimensionaler Eikörper. II. Arch. Math. 4, 374–379 (1953)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Klain, D.A.: Even valuations on convex bodies. Trans. Am. Math. Soc. 352(1), 71–93 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357(10), 4191–4213 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ludwig, M.: Intersection bodies and valuations. Am. J. Math. 128(6), 1409–1428 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ludwig, M.: Minkowski areas and valuations. J. Differ. Geom. 86(1), 133–161 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ludwig, M.: Fisher information and matrix-valued valuations. Adv. Math. 226(3), 2700–2711 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ludwig, M., Reitzner, M.: A classification of ${\rm SL}(n)$ invariant valuations. Ann. Math. 172(2), 1219–1267 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Parapatits, L.: ${\rm SL}(n)$-contravariant $L_p$-Minkowski valuations. Trans. Am. Math. Soc. 366(3), 1195–1211 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Schneider, R.: Equivariant endomorphisms of the space of convex bodies. Trans. Am. Math. Soc. 194, 53–78 (1974)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)Google Scholar
  16. 16.
    Schneider, R., Schuster, F.E.: Rotation equivariant Minkowski valuations. Int. Math. Res. Not. 2006, 72894 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Wannerer, T.: ${\rm GL}(n)$ equivariant Minkowski valuations. Indiana Univ. Math. J. 60(5), 1655–1672 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSuzhou University of Science and TechnologySuzhouChina

Personalised recommendations