Mathematische Annalen

, Volume 347, Issue 3, pp 703–737 | Cite as

Inequalities for mixed p-affine surface area

Article

Abstract

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of L p affine surface areas, mixed p-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.

Mathematics Subject Classification (2000)

52A20 53A15 

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References

  1. 1.
    Aleksandrov A.D.: On the theory of mixed volumes of convex bodies. II: new inequalities between mixed volumes and their applications. Mat. Sb. (N. S.) [Russian] 2, 1205–1238 (1937)MATHGoogle Scholar
  2. 2.
    Alesker S.: Continuous rotation invariant valuations on convex sets. Ann. Math. 149, 977–1005 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alesker S.: Description of translation invariant valuations on convex sets with a solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11, 244–272 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Andrews B.: Contraction of convex hypersurfaces by their affine normal. J. Diff. Geom. 43, 207–230 (1996)MATHGoogle Scholar
  5. 5.
    Andrews B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138, 151–161 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bourgain J., Milman V.D.: New volume ratio properties for convex symmetric bodies in \({\mathbb{R}^{n}}\) . Invent. Math. 88, 319–340 (1987)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Burago Y.D., Zalgaller V.A.: Geometric Inequalities. Springer, Berlin (1988)MATHGoogle Scholar
  8. 8.
    Busemann, H.: Convex Surface, Interscience Tracts in Pure and Appl. Math., No. 6, Interscience, New York (1958). MR 21 #3900Google Scholar
  9. 9.
    Campi S., Gronchi P.: The L p Busemann-Petty centroid inequality. Adv. Math. 167, 128–141 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen W.: L p Minkowski problem with not necessarily positive data. Adv. Math. 201, 77–89 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chou K., Wang X.: The L p-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fleury B., Guédon O., Paouris G.: A stability result for mean width of L p-centroid bodies. Adv. Math. 214, 865–877 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Folland G.B.: Real Analysis. Wiley, New York (1999)MATHGoogle Scholar
  14. 14.
    Gardner R.J.: Geometric Tomography. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  15. 15.
    Gardner R.J., Zhang G.: Affine inequalities and radial mean bodies. Am. J. Math. 120(3), 505–528 (1998)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gruber, P.M.: Aspects of Approximation of Convex Bodies: Handbook of Convex Geometry, vol. A, pp. 321–345. North–Holland, New York (1993)Google Scholar
  17. 17.
    Hardy G.H., Littlewood J.E., Pólya G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)MATHGoogle Scholar
  18. 18.
    Hu C., Ma X., Shen C.: On the Christoffel–Minkowski problem of Fiery’s p-sum. Calc. Var. Partial Diff. Equ. 21(2), 137–155 (2004)MATHMathSciNetGoogle Scholar
  19. 19.
    Hug D.: Curvature relations and affine surface area for a general convex body and its polar. Results Math. V 29, 233–248 (1996)MATHMathSciNetGoogle Scholar
  20. 20.
    Klartag B.: A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kuperberg G.: From the Mahler conjecture to Gauss linking integrals. Geom. Funct. Anal. 18, 870–892 (2008)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Leichtweiss K.: Konvexe Mengen. Springer, Berlin (1980)Google Scholar
  23. 23.
    Ludwig M., Reitzner M.: A characterization of affine surface area. Adv. Math. 147, 138–172 (1999)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. Math. (in press)Google Scholar
  25. 25.
    Ludwig M., Schütt C., Werner E.: Approximation of the Euclidean ball by polytopes. Studia Math. 173, 1–18 (2006)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lutwak E.: Dual mixed volumes. Pac. J. Math. 58, 531–538 (1975)MATHMathSciNetGoogle Scholar
  27. 27.
    Lutwak E.: Mixed affine surface area. J. Math. Anal. Appl. 125, 351–360 (1987)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lutwak E.: The Brunn–Minkowski–Firey theory I: mixed volumes and the Minkowski problem. J. Diff. Geom. 38, 131–150 (1993)MATHMathSciNetGoogle Scholar
  29. 29.
    Lutwak E.: The Brunn–Minkowski–Firey theory. II: affine and geominimal surface areas. Adv. Math. 118(2), 244–294 (1996)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Lutwak E., Oliker V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Diff. Geom. 41, 227–246 (1995)MATHMathSciNetGoogle Scholar
  31. 31.
    Lutwak E., Yang D., Zhang G.: L p affine isoperimetric inequalities. J. Diff. Geom. 56, 111–132 (2000)MATHMathSciNetGoogle Scholar
  32. 32.
    Lutwak E., Yang D., Zhang G.: Sharp affine L p Sobolev inequalities. J. Diff. Geom. 62, 17–38 (2002)MATHMathSciNetGoogle Scholar
  33. 33.
    Lutwak E., Yang D., Zhang G.: On the L p-Minkowski problem. Trans. Am. Math. Soc. 356, 4359–4370 (2004)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Meyer M., Werner E.: The Santaló-regions of a convex body. Trans. Am. Math. Soc. 350(11), 4569–4591 (1998)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Meyer M., Werner E.: On the p-affine surface area. Adv. Math. 152, 288–313 (2000)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Sapiro G., Tannenbaum A.: On affine plane curve evolution. J. Funct. Anal. 119, 79–120 (1994)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Schneider R.: Convex Bodies: The Brunn–Minkowski theory. Cambridge University Press, Cambridge (1993)MATHCrossRefGoogle Scholar
  38. 38.
    Schütt, C., Werner, E.: Random polytopes of points chosen from the boundary of a convex body. In: GAFA Seminar Notes, in Lecture Notes in Math., vol. 1807, pp. 241–422. Springer (2002)Google Scholar
  39. 39.
    Schütt C., Werner E.: Surface bodies and p-affine surface area. Adv. Math. 187, 98–145 (2004)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Stancu A.: The discrete planar L 0-Minkowski problem. Adv. Math. 167, 160–174 (2002)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Stancu A.: On the number of solutions to the discrete two-dimensional L 0-Minkowski problem. Adv. Math. 180, 290–323 (2003)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Trudinger N.S., Wang X.: The affine Plateau problem. J. Am. Math. Soc. 18, 253–289 (2005)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Wang W., Leng G.: L p-mixed affine surface area. J. Math. Anal. Appl. 335, 341–354 (2007)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Wang, X.: Affine maximal hypersurfaces. In: Proceedings of the International Congress of Mathematicians, vol. 3, pp. 221–231. Beijing (2002)Google Scholar
  45. 45.
    Werner E.: Illumination bodies and affine surface area. Studia Math. 110, 257–269 (1994)MATHMathSciNetGoogle Scholar
  46. 46.
    Werner E., Ye D.: New L p-affine isoperimetric inequalities. Adv. Math. 218, 762–780 (2008)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  2. 2.UFR de Mathématique, Université de Lille 1Villeneuve d’AscqFrance

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