f-Divergence for Convex Bodies

  • Elisabeth M. Werner
Part of the Fields Institute Communications book series (FIC, volume 68)


We introduce f-divergence, a concept from information theory and statistics, for convex bodies in ℝ n . We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the L p affine surface area from the L p Brunn Minkowski theory are special cases of f-divergences.

Key words

f-divergence Relative entropy Affine surface area 



Prof. Werner’s work was partially supported by an NSF grant, a FRG-NSF grant and a BSF grant.


  1. 1.
    M.S. Ali, D. Silvey, A general class of coefficients of divergence of one distribution from another. J. Royal Stat. Soc. Ser. B 28, 131–142 (1966)MathSciNetMATHGoogle Scholar
  2. 2.
    I. Csiszár, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Publ. Math. Inst. Hungar. Acad. Sci. ser. A 8, 84–108 (1963)Google Scholar
  3. 3.
    T. Cover, J. Thomas, Elements of Information Theory, 2nd edn. (Wiley, Hoboken, 2006)MATHGoogle Scholar
  4. 4.
    R.J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math. 140(2), 435–47 (1994)MATHCrossRefGoogle Scholar
  5. 5.
    R.J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities. Adv. Math. 216, 358–386 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R.J. Gardner, G. Zhang, Affine inequalities and radial mean bodies. Am. J. Math. 120(3), 505–528 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    E. Grinberg, G. Zhang, Convolutions, transforms, and convex bodies. Proc. London Math. Soc. 78(3), 77–115 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R.J. Gardner, A. Koldobsky, T. Schlumprecht, An analytical solution to the Busemann-Petty problem on sections of convex bodies. Ann. Math. 149(2), 691–703 (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    C. Haberl, Blaschke valuations. Amer. J. Math. 133, 717–751 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    C. Haberl, F. Schuster, General Lp affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem. Adv. Math. 224, 2485–2510 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    D. Hug, Curvature Relations and Affine Surface Area for a General Convex Body and its Polar. Results Math. V. 29, 233–248 (1996)Google Scholar
  13. 13.
    J. Jenkinson, E. Werner Relative entropies for convex bodies, to appear in Transactions of the AMS.Google Scholar
  14. 14.
    D. Klain, Star valuations and dual mixed volumes. Adv. Math. 121, 80–101 (1996)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D. Klain, Invariant valuations on star-shaped sets. Adv. Math. 125, 95–113 (1997)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    F. Liese, I. Vajda, On Divergences and Information in Statistics and Information Theory. IEEE Trans. Inf. Theory 52, 4394–4412 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Lindenstrauss, L. Tzafriri, in Classical Banach spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92 (Springer, Berlin, 1977)Google Scholar
  18. 18.
    M. Ludwig, Ellipsoids and matrix valued valuations. Duke Math. J. 119, 159–188 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    M. Ludwig, Minkowski areas and valuations. J. Differ. Geom. 86, 133–162 (2010)MathSciNetMATHGoogle Scholar
  20. 20.
    M. Ludwig, General affine surface areas. Adv. Math. 224, 2346–2360 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    M. Ludwig, M. Reitzner, A Characterization of Affine Surface Area. Adv. Math. 147, 138–172 (1999)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    M. Ludwig, M. Reitzner, A classification of SL(n) invariant valuations. Ann. Math. 172 ​, 1223–1271 (2010)Google Scholar
  23. 23.
    E. Lutwak, Dual mixed volumes. Pacific J. Math. 58, 531–538 (1975)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    E. Lutwak, Intersection bodies and dual mixed volumes. Adv. Math. 71, 232–261 (1988)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    E. Lutwak, The Brunn-Minkowski-Firey theory I : Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)MathSciNetMATHGoogle Scholar
  26. 26.
    E. Lutwak, The Brunn-Minkowski-Firey theory II : Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    E. Lutwak, G. Zhang, Blaschke-Santaló inequalities. J. Differ. Geom. 47, 1–16 (1997)MathSciNetMATHGoogle Scholar
  28. 28.
    E. Lutwak, D. Yang, G. Zhang, A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    E. Lutwak, D. Yang, G. Zhang, Sharp Affine L p Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)MathSciNetMATHGoogle Scholar
  30. 30.
    E. Lutwak, D. Yang, G. Zhang, The Cramer–Rao inequality for star bodies. Duke Math. J. 112, 59–81 (2002)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    E. Lutwak, D. Yang, G. Zhang, Volume inequalities for subspaces of L p. J. Differ. Geom. 68, 159–184 (2004)MathSciNetMATHGoogle Scholar
  32. 32.
    E. Lutwak, D. Yang, G. Zhang, Moment-entropy inequalities. Ann. Probab. 32, 757–774 (2004)MathSciNetMATHGoogle Scholar
  33. 33.
    E. Lutwak, D. Yang, G. Zhang, Cramer-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information. IEEE Trans. Inf. Theory 51, 473–478 (2005)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    M. Meyer, E. Werner, The Santaló-regions of a convex body. Trans. AMS 350(11), 4569–4591 (1998)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    M. Meyer, E. Werner, On the p-affine surface area. Adv. Math. 152, 288–313 (2000)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    T. Morimoto, Markov processes and the H-theorem. J. Phys. Soc. Jap. 18, 328–331 (1963)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    F. Nazarov, F. Petrov, D. Ryabogin, A. Zvavitch, A remark on the Mahler conjecture: local minimality of the unit cube. Duke Math. J. 154, 419–430 (2010)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    G. Paouris, E. Werner, Relative entropy of cone measures and L p-centroid bodies. Proc. London Math. Soc. (2011). DOI 10.1112/plms/pdr030Google Scholar
  39. 39.
    C. Petty, Affine isoperimetric problems, discrete geometry and convexity. Ann. NY. Acad. Sci. 441, 113–127 (1985)MathSciNetCrossRefGoogle Scholar
  40. 40.
    A. Rényi, On measures of entropy and information, in Proceedings of the 4th Berkeley Symposium on Probability Theory and Mathematical Statistics, vol.1 (University of California Press, California 1961), pp. 547–561Google Scholar
  41. 41.
    B. Rubin, G. Zhang, Generalizations of the Busemann-Petty problem for sections of convex bodies. J. Funct. Anal. 213, 473–501 (2004)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    F. Schuster, Crofton measures and Minkowski valuations. Duke Math. J. 154, 1–30 (2010)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    C. Schütt, On the affine surface area. Proc. Am. Math. Soc. 118, 275–290 (1990)Google Scholar
  44. 44.
    C. Schütt, E. Werner, The convex floating body. Math. Scand. 66, 275–290 (1990)MathSciNetMATHGoogle Scholar
  45. 45.
    C. Schütt, E. Werner, Random polytopes of points chosen from the boundary of a convex body, in GAFA Seminar Notes. Lecture Notes in Mathematics vol. 1807 (Springer-Verlag 2002), 241–422.Google Scholar
  46. 46.
    C. Schütt, E. Werner, Surface bodies and p-affine surface area. Adv. Math. 187, 98–145 (2004)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    A. Stancu, The Discrete Planar L 0-Minkowski Problem. Adv. Math. 167, 160–174 (2002)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    A. Stancu, On the number of solutions to the discrete two-dimensional L 0-Minkowski problem. Adv. Math. 180, 290–323 (2003)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    E. Werner, Illumination bodies and affine surface area. Studia Math. 110, 257–269 (1994)MathSciNetMATHGoogle Scholar
  50. 50.
    E. Werner, On L p-affine surface areas. Indiana Univ. Math. J. 56(5), 2305–2324 (2007)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    E. Werner, Rényi Divergence and L p-affine surface area for convex bodies. Adv. Math. 230, 1040–1059 (2012)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    E. Werner, D. Ye, New L p affine isoperimetric inequalities. Adv. Math. 218(3), 762–780 (2008)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    E. Werner, D. Ye, Inequalities for mixed p-affine surface area. Math. Ann. 347, 703–737 (2010)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    D. Ye, Inequalities for general mixed affine surface areas. J. London Math. Soc. 85, 101–120 (2012)MATHCrossRefGoogle Scholar
  55. 55.
    G. Zhang, Intersection bodies and Busemann-Petty inequalities in ℝ4. Ann. Math. 140, 331–346 (1994)MATHCrossRefGoogle Scholar
  56. 56.
    G. Zhang, A positive answer to the Busemann-Petty problem in four dimensions. Ann. Math. 149, 535–543 (1999)MATHCrossRefGoogle Scholar
  57. 57.
    G. Zhang, New affine isoperimetric inequalities, in International Congress of Chinese Mathematicians (ICCM), Hangzhou, China, vol. II, pp. 239–267 (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  2. 2.UFR de MathématiqueVilleneuve d’AscqFrance

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