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Translative containment measure and symmetric mixed isohomothetic inequalities

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Abstract

We first investigate the translative containment measure for convex domain K 0 to contain, or to be contained in, the homothetic copy of another convex domain K 1, i.e., given two convex domains K 0, K 1 of areas A 0, A 1, respectively, in the Euclidean plane ℝ2, is there a translation T so that t(TK 1) ⊂ K 0 or t(TK 1) ⊃ K 0 for t > 0? Via the translative kinematic formulas of Poincaré and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficit σ2(K 0,K 1) ≡ A 201 − A 0 A 1, where A 01 is the mixed area of K 0 and K 1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(TK 1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.

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References

  1. Aubin T. Problèmes isopérimétriques et espaces de Sobolev. J Differential Geom, 1976; 11: 573–598

    MATH  MathSciNet  Google Scholar 

  2. Banchoff T, Pohl W. A generalization of the isoperimetric inequality. J Differential Geom, 1971; 6: 175–213

    MATH  MathSciNet  Google Scholar 

  3. Blaschke W. Vorlesungen über Intergralgeometrie, 3rd ed. Berlin: Deutsch Verlag Wiss, 1955

    Google Scholar 

  4. Bokowski J, Heil E. Integral representation of quermassintegrals and Bonnesen-style inequalities. Arch Math, 1986; 47: 79–89

    Article  MATH  MathSciNet  Google Scholar 

  5. Bol G. Zur theorie der konvexen Körper (in German). Jahresber Deutsch Math Verein, 1939; 49: 113–123

    MathSciNet  Google Scholar 

  6. Bonnesen T. Les probléms des isopérimétres et des isépiphanes. Paris: Gauthier-Villars, 1920

    Google Scholar 

  7. Böröczky K, Lutwak E, Yang D, et al. The log-Brunn-Minkowski inequality. Adv Math, 2012; 231: 1974–1997

    Article  MATH  MathSciNet  Google Scholar 

  8. Burago Y, Zalgaller V. Geometric Inequalities. Berlin-Heidelberg: Springer-Verlag, 1988

    Book  MATH  Google Scholar 

  9. Croke C. A sharp four-dimensional isoperimetric inequality. Comment Math Helv, 1984; 59: 187–192

    Article  MATH  MathSciNet  Google Scholar 

  10. Diskant V. A generalization of Bonnesen’s inequalities. Soviet Math Dokl, 1973; 14: 1728–1731

    MATH  Google Scholar 

  11. Flanders H. A proof of Minkowski’s inequality for convex curves. Amer Math Monthly, 1968; 75: 581–593

    Article  MATH  MathSciNet  Google Scholar 

  12. Fujiwara M. Ein Satz über knovexe geschlossene Kurven. Sci Rep Töhoku Univ, 1920; 19: 289–294

    Google Scholar 

  13. Fusco N, Maggi F, Pratelli A. The sharp quantitative isoperimetric inequality. Ann of Math, 2008; 168: 941–980

    Article  MATH  MathSciNet  Google Scholar 

  14. Gage M. An isoperimetric inequality with applications to curve shortening. Duke Math J, 1983; 50: 1225–1229

    Article  MATH  MathSciNet  Google Scholar 

  15. Gao X. A note on the reverse isoperimetric inequality. Results Math, 2011; 59: 83–90

    Article  MATH  MathSciNet  Google Scholar 

  16. Gao X. A new reverse isoperimetric inequality and its stability. Math Inequal Appl, 2012; 15: 733–743

    MATH  MathSciNet  Google Scholar 

  17. Green M, Osher S. Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J Math, 1999; 3: 659–676

    MATH  MathSciNet  Google Scholar 

  18. Howard R. The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces. Proc Amer Math Soc, 1998; 126: 2779–2787

    Article  MATH  MathSciNet  Google Scholar 

  19. Howard R, Treibergs A. A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature. Rocky Mountain J Math, 1995; 25: 635–684

    Article  MATH  MathSciNet  Google Scholar 

  20. Hsiung C. Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary. Ann of Math, 1961; 73: 213–220

    Article  MATH  MathSciNet  Google Scholar 

  21. Klain D. Bonnesen-type inequalities for surfaces of constant curvature. Adv Appl Math, 2007; 39: 143–154

    Article  MATH  MathSciNet  Google Scholar 

  22. Klain D, Rota G-C. Introduction to Geometric Probability. Cambridge: Cambridge University Press, 1997

    MATH  Google Scholar 

  23. Kotlyar B. On a geometric inequality. J Soviet Math, 1990; 51: 2534–2536

    Article  MATH  MathSciNet  Google Scholar 

  24. Li M, Zhou J. An upper limit for the isoperimetric deficit of convex set in a plane of constant curvature. Sci China Math, 2010; 53: 1941–1946

    Article  MATH  MathSciNet  Google Scholar 

  25. Osserman R. The isoperimetric inequality. Bull Amer Math Soc, 1978; 84: 1182–1238

    Article  MATH  MathSciNet  Google Scholar 

  26. Osserman R. Bonnesen-style isoperimetric inequality. Amer Math Monthly, 1979; 86: 1–29

    Article  MATH  MathSciNet  Google Scholar 

  27. Pan S, Zhang H. A reverse isoperimetric inequality for closed strictly convex plane curves. Beiträge Algebra Geom, 2007; 48: 303–308

    MATH  MathSciNet  Google Scholar 

  28. Pleijel A. On konvexa kurvor. Normat, 1955; 3: 57–64

    MATH  MathSciNet  Google Scholar 

  29. Polya G, Szego G. Isoperimetric Inequalities in Mathematical Physics. Princeton: Princeton University Press, 1951

    MATH  Google Scholar 

  30. Sangwine-Yager J. A Bonnesen-style inradius inequality in 3-space. Pacific J Math, 1988; 134: 173–178

    Article  MATH  MathSciNet  Google Scholar 

  31. Sangwine-Yager J. Mixed volumes. In: Gruber P, Wills J, eds. Handbook of Convex Geometry, vol. A. Amsterdam: North-Holland, 1993, 43–71

    Chapter  Google Scholar 

  32. Santaló L. Integral Geometry and Geometric Probability. Reading: Addison-Wesley, 1976

    MATH  Google Scholar 

  33. Schneider R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014

    Google Scholar 

  34. Stone A. On the isoperimetric inequality on a minimal surface. Calc Var Partial Differential Equations, 2003; 17: 369–391

    Article  MATH  MathSciNet  Google Scholar 

  35. Teufel E. A generalization of the isoperimetric inequality in the hyperbolic plane. Arch Math, 1991; 57: 508–513

    Article  MATH  MathSciNet  Google Scholar 

  36. Teufel E. Isoperimetric inequalities for closed curves in spaces of constant curvature. Results Math, 1992; 22: 622–630

    Article  MATH  MathSciNet  Google Scholar 

  37. Wei S, Zhu M. Sharp isoperimetric inequalities and sphere theorems. Pacific J Math, 2005; 220: 183–195

    Article  MATH  MathSciNet  Google Scholar 

  38. Weiner J. A generalization of the isoperimetric inequality on the 2-sphere. Indiana Univ Math J, 1974; 24: 243–248

    Article  MATH  MathSciNet  Google Scholar 

  39. Weiner J. Isoperimetric inequalities for immersed closed spherical curves. Proc Amer Math Soc, 1994; 120: 501–506

    Article  MATH  MathSciNet  Google Scholar 

  40. Xia Y, Xu W, Zhou J, et al. Reverse Bonnesen style inequalities in a surface X2ɛ of constant curvature. Sci China Math, 2013; 56: 1145–1154

    Article  MATH  MathSciNet  Google Scholar 

  41. Xu W, Zhou J, Zhu B. On containment measure and the mixed isoperimetric inequality. J Inequal Appl, 2013; 11: 1–11

    Article  MATH  MathSciNet  Google Scholar 

  42. Xu W, Zhou J, Zhu B. Bonnesen-style symmetric mixed isoperimetric inequality. Springer Proc Math Statist, 2014; 106: 97–108

    Article  MathSciNet  Google Scholar 

  43. Yau S. Isoperimetric constants and the first eigenvalue of a compact manifold. Ann Sci Ec Norm Super, 1975; 8: 487–507

    MATH  Google Scholar 

  44. Zeng C, Ma L, Zhou J, et al. The Bonnesen isoperimetric inequality in a surface of constant curvature. Sci China Math, 2012; 55: 1913–1919

    Article  MATH  MathSciNet  Google Scholar 

  45. Zeng C, Zhou J, Yue S. The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math Sinica, 2012; 55: 355–362

    MATH  MathSciNet  Google Scholar 

  46. Zhang G, Zhou J. Containment measures in integral geometry. In: Integral Geometry and Convexity. Singapore: World Scientific, 2006, 153–168

    Chapter  Google Scholar 

  47. Zhang X-M. Schur-convex functions and isoperimetric inequalities. Proc Amer Math Soc, 1998; 126: 461–470

    Article  MATH  MathSciNet  Google Scholar 

  48. Zhou J. On Bonnesen-type inequalities. Acta Math Sinica, 2007; 50: 1397–1402

    MATH  Google Scholar 

  49. Zhou J, Chen F. The Bonnesen-type inequality in a plane of constant cuvature. J Korean Math Soc, 2007; 44: 1363–1372

    Article  MATH  MathSciNet  Google Scholar 

  50. Zhou J, Ren D. Geometric inequalities—from integral geometry point of view (in Chinese). Acta Math Sci Ser A Chin Ed, 2010; 30: 1322–1339

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Miao Luo, WenXue Xu & JiaZu Zhou

  2. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China

    Miao Luo

  3. School of Mathematical Sciences, Kaili University, Kaili, 556011, China

    JiaZu Zhou

  4. Department of Computer Science, Southeast Guizhou Vocational College of Technology for Nationalities, Kaili, 556000, China

    JiaZu Zhou

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  1. Miao Luo
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  2. WenXue Xu
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  3. JiaZu Zhou
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Correspondence to JiaZu Zhou.

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Luo, M., Xu, W. & Zhou, J. Translative containment measure and symmetric mixed isohomothetic inequalities. Sci. China Math. 58, 2593–2610 (2015). https://doi.org/10.1007/s11425-015-5074-5

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  • DOI: https://doi.org/10.1007/s11425-015-5074-5

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