Advertisement

SL(n) Contravariant \(L_{p}\) Harmonic Valuations on Polytopes

  • Lijuan Liu
  • Wei WangEmail author
Article

Abstract

All SL(n) contravariant \(L_{p}\) harmonic valuations on convex polytopes are completely classified without homogeneity assumptions.

Keywords

Convex ploytope SL(n) contravariance Valuation 

Mathematics Subject Classification

52B45 52A20 

Notes

Acknowledgements

The work of the first author was supported by China Scholarship Council (CSC 201808430267) and the Natural Science Foundation of Hunan Province (2019JJ50172). The work of the second author was supported by the Natural Science Foundation of Hunan Province (2017JJ3085).

References

  1. 1.
    Alesker, S.: Continuous rotation invariant valuations on convex sets. Ann. Math. 149(3), 977–1005 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alesker, S.: Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11(2), 244–272 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alesker, S., Bernig, A., Schuster, F.E.: Harmonic analysis of translation invariant valuations. Geom. Funct. Anal. 21(4), 751–773 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Campi, S., Gronchi, P.: The \(L^{p}\)-Busemann–Petty centroid inequality. Adv. Math. 167(1), 128–141 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fleury, B., Guedon, O., Paouris, G.: A stability result for mean width of \(L_{p}\)-centroid bodies. Adv. Math. 214(2), 865–877 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gardner, R.J.: A positive answer to the Busemann–Petty problem in three dimensions. Ann. Math. 140(2), 435–447 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gardner, R.J., Giannopoulos, A.A.: \(p\)-cross-section bodies. Indiana Univ. Math. J. 48(2), 593–613 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gardner, R.J., Koldobsky, A., Schlumprecht, T.: An analytic solution to the Busemann–Petty problem on sections of convex bodies. Ann. Math. 149(2), 691–703 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Grinberg, E., Zhang, G.: Convolutions, transforms, and convex bodies. Proc. Lond. Math. Soc. 78(3), 77–115 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007)Google Scholar
  11. 11.
    Haberl, C.: Star body valued valuations. Indiana Univ. Math. J. 58(5), 2253–2267 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Haberl, C.: Blaschke valuations. Am. J. Math. 133(3), 717–751 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Haberl, C.: Minkowski valuations intertwining the special linear group. J. Eur. Math. Soc. (JEMS) 14(5), 1565–1597 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Haberl, C., Ludwig, M.: A characterization of \(L_{p}\) intersection bodies. Int. Math. Res. Not. 2006, 10548 (2006)zbMATHGoogle Scholar
  15. 15.
    Haberl, C., Parapatits, L.: The centro-affine Hadwiger theorem. J. Am. Math. Soc. 27(3), 685–705 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Haberl, C., Parapatits, L.: Valuations and surface area measures. J. Reine Angew. Math. 687, 225–245 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Haberl, C., Parapatits, L.: Moments and valuations. Am. J. Math. 138(6), 1575–1603 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Haberl, C., Parapatits, L.: Centro-affine tensor valuations. Adv. Math. 316, 806–865 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin (1957)zbMATHCrossRefGoogle Scholar
  20. 20.
    Klain, D.A.: Star valuations and dual mixed volumes. Adv. Math. 121(1), 80–101 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Klain, D.A., Rota, G.-C.: Introduction to Geometric Probability. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  22. 22.
    Klartag, B., Milman, E.: Centroid bodies and the logarithmic Laplace transform—A unified approach. Geom. Funct. Anal. 262(1), 10–34 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Li, J., Leng, G.: \(L_{p}\) Minkowski valuations on polytopes. Adv. Math. 299, 139–173 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Li, J., Ma, D.: Laplace transforms and valuations. J. Funct. Anal. 272(2), 738–758 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ludwig, M.: Moment vectors of polytopes. Rend. Circ. Mat. Palermo 2(Suppl. 70), 123–138 (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ludwig, M.: Projection bodies and valuations. Adv. Math. 172(2), 158–168 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Ludwig, M.: Valuations on polytopes containing the origin in their interiors. Adv. Math. 170(2), 239–256 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ludwig, M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119(1), 159–188 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357(10), 4191–4213 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ludwig, M.: Intersection bodies and valuations. Am. J. Math. 128(6), 1409–1428 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ludwig, M.: Minkowski areas and valuations. J. Differ. Geom. 86(1), 133–161 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ludwig, M., Reitzner, M.: A classification of \(\rm SL(n)\) invariant valuations. Ann. Math. 172(2), 1219–1267 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ludwig, M., Reitzner, M.: \( {SL}(n)\) invariant valuations on polytopes. Discrete Comput. Geom. 57(3), 571–581 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71(2), 232–261 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Lutwak, E.: Centroid bodies and dual mixed volumes. Proc. Lond. Math. Soc. 60(2), 365–391 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Lutwak, E., Yang, D., Zhang, G.: \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 56(1), 111–132 (2000)zbMATHCrossRefGoogle Scholar
  37. 37.
    Lutwak, E., Yang, D., Zhang, G.: Moment-entropy inequalities. Ann. Probab. 32(1B), 757–774 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Lutwak, E., Zhang, G.: Blaschke–Santaló inequalities. J. Differ. Geom. 47(1), 1–16 (1997)zbMATHCrossRefGoogle Scholar
  39. 39.
    Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \(n\)-dimensional space. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1376, pp. 64–104. Springer, Berlin (1989)CrossRefGoogle Scholar
  40. 40.
    Paouris, G.: Concentration of mass on isotropic convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Parapatits, L.: \( {SL}(n)\)-contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366(3), 1195–1211 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Parapatits, L.: \( {SL}(n)\)-covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89(2), 397–414 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Petty, C.M.: Centroid surfaces. Pac. J. Math. 11, 1535–1547 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Rubin, B.: Generalized Minkowski-Funk transforms and small denominators on the sphere. Fract. Calc. Appl. Anal. 3(2), 177–203 (2000)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  46. 46.
    Schuster, F.E., Wannerer, T.: \( {GL}(n)\) contravariant Minkowski valuations. Trans. Am. Math. Soc. 364(2), 815–826 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Wannerer, T.: \( {GL}(n)\) equivariant Minkowski valuations. Indiana Univ. Math. J. 60(5), 1655–1672 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Yaskin, V., Yaskina, M.: Centroid bodies and comparison of volumes. Indiana Univ. Math. J. 55(3), 1175–1194 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Zeng, C., Ma, D.: \( {SL}(n)\) covariant vector valuations on polytopes. Trans. Am. Math. Soc. 370(12), 8999–9023 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Zhang, G.: A positive solution to the Busemann–Petty problem in \({\mathbb{R}}^{4}\). Ann. Math. 149(2), 535–543 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceHunan University of Science and TechnologyXiangtanPeople’s Republic of China

Personalised recommendations