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The Journal of Geometric Analysis

, Volume 24, Issue 4, pp 1898–1911 | Cite as

On the Stability of the p-Affine Isoperimetric Inequality

  • Mohammad N. IvakiEmail author
Article

Abstract

Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in ℝ2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in ℝ2 such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.

Keywords

Affine support function Affine normal flow Hausdorff distance Stability of geometric inequalities p-affine surface area p-affine isoperimetric inequality 

Mathematics Subject Classification (2010)

52A40 53C44 53A04 52A10 53A15 53A15 

Notes

Acknowledgements

I would like to thank Alina Stancu and Károly Böröczky for comments and suggestions that have improved the initial manuscript. I am indebted to two referees for the very careful reading of the original submission.

References

  1. 1.
    Andrews, B.: Contraction of convex hypersurfaces by their affine normal. J. Differ. Geom. 43, 207–230 (1996) zbMATHGoogle Scholar
  2. 2.
    Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pacific J. Math. 195(1), 1–34 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, B.: Singularities in crystalline curvature flows. Asian J. Math. 6, 101–121 (2002) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ball, K., Böröczky, K.J.: Stability of the Prékopa–Leindler inequality. Mathematika 56, 339–356 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ball, K., Böröczky, K.J.: Stability of some versions of the Prékopa–Leindler inequality. Monatshefte Math. 163, 1–14 (2011) CrossRefzbMATHGoogle Scholar
  6. 6.
    Barthe, F., Böröczky, K.J., Fradelizi, M.: Stability of the functional forms of the Blaschke–Santaló inequality. arXiv:1206.0369v1 [math.MG]
  7. 7.
    Blaschke, W.: Über affine Geometrie I. Isoperimetrische Eigenschaften von Ellipse und Ellipsoid. Leipz. Ber. 68, 217–239 (1916) Google Scholar
  8. 8.
    Böröczky, K.J., Hug, D.: Stability of the reverse Blaschke–Santaló inequality for zonoids and applications. Adv. Appl. Math. 44, 309–328 (2010) CrossRefzbMATHGoogle Scholar
  9. 9.
    Böröczky, K.J.: The stability of the Rogers–Shephard inequality. Adv. Math. 190, 47–76 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Böröczky, K.J.: Stability of Blaschke–Santaló inequality and the affine isoperimetric inequality. Adv. Math. 225, 1914–1928 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Böröczky, K.J., Makai, E. Jr.: On the volume product of planar polar convex bodies-upper estimates: the polygonal case and stability. In preparation Google Scholar
  12. 12.
    Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. PDEs. 36, 419–436 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diskant, V.I.: Stability of the solution of a Minkowski equation. Sib. Mat. Zh. 14, 669–673 (1973) (in Russian). Engl. transl.: Siberian Math. J. 14, 466–473 (1974) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hug, D.: Contributions to affine surface area. Manuscr. Math. 91, 283–301 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hug, D., Schneider, R.: A stability result for volume ratio. Isr. J. Math. 161, 209–219 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Groemer, H.: Stability of geometric inequalities. In: Handbook of Convex Geometry, pp. 125–150. North-Holland, Amsterdam (1993) CrossRefGoogle Scholar
  17. 17.
    Gruber, P.M.: Asymptotic estimates for best and stepwise approximation of convex bodies II. Forum Math. 5, 521–538 (1993) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007) zbMATHGoogle Scholar
  19. 19.
    Ivaki, M.N.: Centro-affine curvature flows on centrally symmetric convex curves. Trans. Am. Math. Soc., to appear. arXiv:1205.6456v2 [math.DG]
  20. 20.
    Ivaki, M.N.: A flow approach to the L −2 Minkowski problem. Adv. Appl. Math. 50, 445–464 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on His 60th Birthday, Jan. 8, 1948, pp. 187–204. Interscience, New York (1948) Google Scholar
  22. 22.
    Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. Math. 172, 1223–1271 (2010) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ludwig, M.: General affine surface area. Adv. Math. 224, 2346–2360 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lutwak, E., Oliker, V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227–246 (1995) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lutwak, E.: The Brunn–Minkowski–Firey theory. I: Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Lutwak, E.: The Brunn–Minkowski–Firey theory. II: Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the L p-Minkowski problem. Int. Math. Res. Notices 2006 (2006). doi: 10.1155/IMRN/2006/62987
  28. 28.
    Lutwak, E., Yang, D., Zhang, G.: Sharp affine L p Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lutwak, E., Yang, D., Zhang, G.: The Cramer–Rao inequality for star bodies. Duke Math. J. 112, 59–81 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lutwak, E., Yang, D., Zhang, G.: L p affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Meyer, M., Werner, E.: On the p-affine surface area. Adv. Math. 152, 288–313 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Petty, C.M.: Affine isoperimetric problems, discrete geometry and convexity. In: Goodman, J.E., Lutwak, E., Malkevitch, J., Pollack, R. (eds.) Proc. Conf. New York 1982. Annals New York Acad. Sci., vol. 440, pp. 113–127 (1985) Google Scholar
  33. 33.
    Schütt, C., Werner, E.: Surface bodies and p-affine surface area. Adv. Math. 187, 98–145 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sapiro, G., Tannenbaum, A.: On affine plane curve evolution. J. Funct. Anal. 119, 79–120 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Stancu, A.: The discrete planar L 0-Minkowski problem. Adv. Math. 167, 160–174 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Stancu, A.: On the number of solutions to the discrete two-dimensional L 0-Minkowski problem. Adv. Math. 180, 290–323 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Stancu, A.: Two volume product inequalities and their applications. Can. Math. Bull. 52, 464–472 (2004) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Stancu, A.: The floating body problem. Bull. Lond. Math. Soc. 38, 839–846 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stancu, A.: The necessary condition for the discrete L 0-Minkowski problem in ℝ2. J. Geom. 88, 162–168 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Stancu, A.: Centro-affine invariants for smooth convex bodies. Int. Math. Res. Not. (2011). doi: 10.1093/imrn/rnr110 Google Scholar
  41. 41.
    Stancu, A.: Some affine invariants revisited. arXiv:1208.0783v1 [math.FA]
  42. 42.
    Trudinger, N.S., Wang, X.J.: The Bernstein problem for affine maximal hypersurfaces. Invent. Math. 140, 399–422 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Trudinger, N.S., Wang, X.J.: Affine complete locally convex hypersurfaces. Invent. Math. 150, 45–60 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Trudinger, N.S., Wang, X.J.: The affine Plateau problem. J. Am. Math. Soc. 18, 253–289 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Trudinger, N.S., Wang, X.J.: Boundary regularity for the Monge–Ampère and affine maximal surface equations. Ann. Math. 167, 993–1028 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Werner, E., Ye, D.: New L p affine isoperimetric inequalities. Adv. Math. 218, 762–780 (2008) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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