Geometric and Functional Analysis

, Volume 27, Issue 3, pp 466–488 | Cite as

Minkowski Endomorphisms

  • Felix DorrekEmail author
Open Access


Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not weakly-monotone. This answers questions posed repeatedly by Kiderlen (Trans Am Math Soc 358:5539–5564, 2006), Schneider (Convex bodies: the Brunn–Minkowski theory. Second expanded edition, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge, 2014) and Schuster (Trans Am Math Soc 359:5567–5591, 2007). Furthermore, a recent representation result for Minkowski valuations by Schuster and Wannerer is improved under additional homogeneity assumptions. Also a question related to the structure of Minkowski endomorphisms by the same authors is answered. Finally, it is shown that there exists no McMullen decomposition in the class of continuous, even, SO(n)-equivariant and translation invariant Minkowski valuations extending a result by Parapatits and Wannerer (Duke Math J 162:1895–1922, 2013).



Open access funding provided by Austrian Science Fund (FWF). The work of the author was supported by the European Research Council (ERC), Project Number: 306445, and the Austrian Science Fund (FWF), Project Number: Y603-N26.


  1. Aba12.
    Abardia J.: Difference bodies in complex vector spaces. Journal of Functional Analysis 263, 3588–3603 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. AB11.
    Abardia J., Bernig A.: Projection bodies in complex vector spaces. Advances in Mathematics 227, 830–846 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ale01.
    Alesker S.: Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geometric and Functional Analysis 11, 244–272 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ale03.
    Alesker S.: Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. Journal of Differential Geometry 63, 63–95 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ale04.
    Alesker S.: The multiplicative structure on continuous polynomial valuations. Geometric and Functional Analysis 14, 1–26 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ale11.
    Alesker S.: A Fourier-type transform on translation-invariant valuations on convex sets. Israel Journal of Mathematics 181, 189–294 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. AM10.
    Artstein-Avidan S., Milman V.: A characterization of the support map. Advances in Mathematics 223(1), 379–391 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ber69.
    C. Berg. Corps convexes et potentiels sphériques. Mat.-Fys. Medd. Danske Vid. Selsk., 37 (1969).Google Scholar
  9. BPSW.
    A. Berg, L. Parapatits, F.E. Schuster, and M. Weberndorfer. Log-Concavity Properties of Minkowski Valuations, preprint.Google Scholar
  10. BF11.
    Bernig A., Fu J.H.G.: Hermitian integral geometry. Annals of Mathematics 173, 907–945 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ber12.
    A. Bernig. Algebraic Integral Geometry, Global Differential Geometry, Springer Proceedings in Mathematics, Vol. 17. Springer, Heidelberg (2012), pp. 107–145.Google Scholar
  12. Fir68.
    Firey W.J.: Christoffel’s problem for general convex bodies. Mathematika 15, 7–21 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Fir70a.
    Firey W.J.: Local behaviour of area functions of convex bodies. Pacific Journal of Mathematics 35, 345–357 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fir70b.
    Firey W.J.: Intermediate Christoffel–Minkowski problems for figures of revolution. Israel Journal of Mathematics 8, 384–390 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. GZ98.
    Goodey P., Zhang G.: Inequalities between projection functions of convex bodies. American Journal of Mathematics 120, 345–367 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gro96.
    H. Groemer. Geometric applications of Fourier series and spherical harmonics, Encyclopedia of Mathematics and Its Applications, Vol. 61. Cambridge University Press, Cambridge (1996).Google Scholar
  17. Hab12.
    Haberl C.: Minkowski valuations intertwining the special linear group. Journal of the European Mathematical Society 14, 1565–1597 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. HP14.
    Haberl C., Parapatits L.: The centro-affine Hadwiger theorem. Journal of the American Mathematical Society 27(3), 685–705 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Had57.
    H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (German). Springer, New York (1957).Google Scholar
  20. Kid06.
    Kiderlen M.: Blaschke- and Minkowski-endomorphisms of convex bodies. Transactions of the American Mathematical Society 358, 5539–5564 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kla00.
    Klain D.A.: Even valuations on convex bodies. Transactions of the American Mathematical Society 352, 71–93 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. KR97.
    D.A. Klain and G.-C. Rota. Introduction to Geometric Probability. Cambridge University Press, Cambridge (1997).Google Scholar
  23. Lud02.
    Ludwig M.: Projection bodies and valuations. Advances in Mathematics 172, 158–168 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lud05.
    Ludwig M.: Minkowski valuations. Transactions of the American Mathematical Society 357, 4191–4213 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lud10.
    Ludwig M.: Minkowski areas and valuations. Journal of Differential Geometry 86, 133–161 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. PS12.
    Parapatits L., Schuster F.E.: The Steiner formula for Minkowski valuations. Advances in Mathematics 230, 978–994 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. PW13.
    Parapatits L., Wannerer T.: On the inverse Klain map. Duke Mathematical Journal 162, 1895–1922 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Rud91.
    W. Rudin, Functional Analysis. International Series in Pure and Applied Mathematics, Second edition. McGraw-Hill, New York (1991).Google Scholar
  29. Sch74a.
    Schneider R.: Equivariant endomorphisms of the space of convex bodies. Transactions of the American Mathematical Society 194, 53–78 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Sch74b.
    Schneider R.: Bewegungsäquivariante, additive und stetige Transformationen kovexer Bereiche. Archiv der Mathematik 25, 303–312 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Sch74c.
    Schneider R.: Additive Transformationen konvexer Körper. Geometriae Dedicata 3, 221–228 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sch14.
    R. Schneider, Convex Bodies: The Brunn–Minkowski Theory. Second Expanded Edition, Encyclopedia of Mathematics and its Applications, Vol. 151. Cambridge University Press, Cambridge (2014).Google Scholar
  33. SS06.
    R. Schneider and F.E. Schuster. Rotation equivariant Minkowski valuations. International Mathematics Research Notices (2006), Art. ID 72894, 20.Google Scholar
  34. Sch07.
    Schuster F.E.: Convolutions and multiplier transformations of convex bodies. Transactions of the American Mathematical Society 359, 5567–5591 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sch10.
    Schuster F.E.: Crofton measures and Minkowski valuations. Duke Mathematical Journal 154, 1–30 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. SW12.
    Schuster F.E., Wannerer T.: GL(n) contravariant Minkowski valuations. Transactions of the American Mathematical Society 364, 815–826 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. SW15.
    Schuster F.E., Wannerer T.: Even Minkowski valuations. American Journal of Mathematics 137, 1651–1683 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. SW.
    F.E. Schuster and T. Wannerer. Minkowski valuations and generalized valuations. arXiv:1507.05412.
  39. Spi76.
    W. Spiegel. Zur Minkowski-Additivität bestimmter Eikörperabbildungen. Journal für die reine und angewandte Mathematik, 286/287 (1976), 164–168.Google Scholar
  40. Tak94.
    M. Takeuchi. Modern spherical functions, Transl. Math. Monogr. 135, Amer. Math. Soc., Providence, RI, 1994.Google Scholar
  41. Wan11.
    Wannerer T.: GL(n) equivariant Minkowski valuations. Indiana University Mathematics Journal 60, 1655–1672 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Wei82.
    Weil W.: Zonoide und verwandte Klassen konvexer Körper. Monatshefte für Mathematik 92(1), 73–84 (1982)CrossRefzbMATHGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Vienna University of Technology, Institute of Discrete Mathematics and GeometryViennaAustria

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