Complex affine isoperimetric inequalities

  • Christoph HaberlEmail author


Complex extensions of the Petty projection inequality and the Busemann–Petty centroid inequality are established.

Mathematics Subject Classification

52A40 52A20 



  1. 1.
    Abardia, J.: Difference bodies in complex vector spaces. J. Funct. Anal. 263(11), 3588–3603 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abardia, J.: Minkowski valuations in a 2-dimensional complex vector space. Int. Math. Res. Not. 5, 1247–1262 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abardia, J., Bernig, A.: Projection bodies in complex vector spaces. Adv. Math. 227, 830–846 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abardia, J., Saorín Gómez, E.: How do difference bodies in complex vector spaces look like? A geometrical approach. Commun. Contemp. Math. 17(4), 145002 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abardia, J., Wannerer, T.: Aleksandrov–Fenchel inequalities for unitary valuations of degree 2 and 3. Calc. Var. Partial Differ. Equ. 54(2), 1767–1791 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Abardia-Evéquoz, J., Böröczky, K., Domokos, M., Kertész, D.: \(SL(m, \mathbb{C})\)-equivariant and translation covariant continuous tensor valuations. (preprint)
  7. 7.
    Bernig, A.: A Hadwiger-type theorem for the special unitary group. Geom. Funct. Anal. 19, 356–372 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bernig, A., Fu, J.H.G., Solanes, G.: Integral geometry of complex space forms. Geom. Funct. Anal. 24, 403–492 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bernig, A., Fu, J.H.G.: Hermitian integral geometry. Ann. Math. 2(173), 907–945 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bolker, E.D.: A class of convex bodies. Trans. Am. Math. Soc. 145, 323–345 (1969)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bourgain, J., Lindenstrauss, J.: Projection bodies. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1317, pp. 250–270. Springer, Berlin, Heidelberg (1988)CrossRefGoogle Scholar
  12. 12.
    Campi, S., Gronchi, P.: The \(L^{p}\)-Busemann–Petty centroid inequality. Adv. Math. 167, 128–141 (2002)Google Scholar
  13. 13.
    Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. Partial Differ. Equ. 36, 419–436 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gardner, R.J.: Geometric Tomography. Encyclopedia of Mathematics and its Applications, vol. 58, 2nd edn. Cambridge University Press, Cambridge (2006)Google Scholar
  15. 15.
    Gruber, P.: Convex and Discrete Geometry. Springer, Berlin (2007)zbMATHGoogle Scholar
  16. 16.
    Haberl, C.: Minkowski valuations intertwining the special linear group. J. Eur. Math. Soc. 14, 1565–1597 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Haberl, C., Schuster, F.: Affine vs. Euclidean isoperimetric inequalities. (preprint)
  18. 18.
    Haberl, C., Schuster, F.: General \({L}_p\) affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2009)Google Scholar
  19. 19.
    Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352, 517–542 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Haddad, J., Jiménez, C.H., Montenegro, M.: Sharp affine Sobolev type inequalities via the \({L}_{p}\) Busemann–Petty centroid inequality. J. Funct. Anal. 271, 454–473 (2016)Google Scholar
  21. 21.
    Koldobsky, A., Paouris, G.G., Zymonopoulou, M.: Complex intersection bodies. J. Lond. Math. Soc. 88(2), 538–562 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Koldobsky, A., König, H., Zymonopoulou, M.: The complex Busemann–Petty problem on sections of convex bodies. Adv. Math. 218(2), 352–367 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Koldobsky, A.: Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs, vol. 116. American Mathematical Society, Providence, RI (2005)CrossRefGoogle Scholar
  24. 24.
    Liu, L., Wang, W., Huang, Q.: On polars of mixed complex projection bodies. Bull. Korean Math. Soc. 52(2), 453–465 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ludwig, M.: Projection bodies and valuations. Adv. Math. 172, 158–168 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191–4213 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lutwak, E.: A general isepiphanic inequality. Proc. Am. Math. Soc. 90(3), 415–421 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lutwak, E.: On some affine isoperimetric inequalities. J. Differ. Geom. 23, 1–13 (1986)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lutwak, E., Yang, D., Zhang, G.: \({L}_p\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000)Google Scholar
  30. 30.
    Lutwak, E., Yang, D., Zhang, G.: Moment-entropy inequalities. Ann. Probab. 32(1B), 757–774 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lutwak, E., Yang, D., Zhang, G.: Orlicz centroid bodies. J. Differ. Geom. 84, 365–387 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lutwak, E., Yang, D., Zhang, G.: Orlicz projection bodies. Adv. Math. 223, 220–242 (2010)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Nguyen, V.H.: New approach to the affine Polya–Szegö principle and the stability version of the affine sobolev inequality. Adv. Math. 302, 1080–1110 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Petty, C.M.: Centroid surfaces. Pac. J. Math. 11, 1535–1547 (1961)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry, pp. 26–41, University of Oklahoma, Norman, OK (1971)Google Scholar
  36. 36.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  37. 37.
    Steineder, C.: Subword complexity and projection bodies. Adv. Math. 217, 2377–2400 (2008)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Thompson, A.C.: Minkowski Geometry. Encyclopedia of Mathematics and its Applications, vol. 63. Cambridge University Press, Cambridge (1996)Google Scholar
  39. 39.
    Wang, T.: The affine Sobolev–Zhang in \(BV(\mathbb{R}^n)\). Adv. Math. 230, 2457–2473 (2012)Google Scholar
  40. 40.
    Wang, W., He, R.G.: Inequalities for mixed complex projection bodies. Taiwan. J. Math. 17(6), 1887–1899 (2013)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wannerer, T.: Integral geometry of unitary area measures. Adv. Math. 263, 1–44 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wannerer, T.: The module of unitarily invariant area measures. J. Differ. Geom. 96, 141–182 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

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