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Bonnesen-Style Symmetric Mixed Isoperimetric Inequality

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Real and Complex Submanifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 106))

Abstract

For convex domains K i  (i = 0, 1) (compact convex sets with non-empty interiors) in the Euclidean plane R 2. Denote by A i and P i areas and circum-perimeters, respectively. The symmetric mixed isoperimetric deficit is \(\varDelta (K_{0},K_{1}):= P_{0}^{2}P_{1}^{2} - 16\pi ^{2}A_{0}A_{1}\). In this paper, we give some Bonnesen-style symmetric mixed inequalities, that is, inequalities of the form \(\varDelta (K_{0},K_{1}) \geq B_{K_{0},K_{1}}\), where \(B_{K_{0},K_{1}}\) is a non-negative invariant of geometric significance and vanishes if and only if both K 0 and K 1 are discs. We also obtain some reverse Bonnesen-style symmetric mixed inequalities. Those inequalities are natural generalizations of known geometric inequalities, such as the known classical isoperimetric inequality.

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References

  1. Banchoff, T.F., Pohl, W.F.: A generalization of the isoperimetric inequality. J. Differ. Geom. 6, 175–213 (1971)

    MATH  MathSciNet  Google Scholar 

  2. Blaschke, W.: Vorlesungen über Intergralgeometrie. 3rd edn. Deutsch, Berlin (1955)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonnesen, T.: Les probléms des isopérimétres et des isépiphanes, Paris (1929)

    Google Scholar 

  5. Bonnesen, T., Fenchel, W.: Theorie der konvexen Köeper, 2nd edn. (1974). Springer, Berlin (1934)

    Google Scholar 

  6. Bottema, O.: Eine obere Grenze für das isoperimetrische Defizit ebener Kurven. Nederl. Akad. Wetensch. Proc. A66, 442–446 (1933)

    Google Scholar 

  7. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  8. Croke, C.: A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv. 59(2), 187–192 (1984)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. Enomoto, K.: A generalization of the isoperimetric inequality on S 2 and flat tori in S 3. Proc. Am. Math. Soc. 120(2), 553–558 (1994)

    MATH  MathSciNet  Google Scholar 

  11. Flanders, H.: A proof of Minkowski’s inequality for convex curves. Am. Math. Mon. 75, 581–593 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grinberg, E., Li, S., Zhang G., Zhou, J.: Integral geometry and convexity. In: Proceedings of the International Conference. World Scientific, Wuhan (2006)

    MATH  Google Scholar 

  13. Gysin, L.: The isoperimetric inequality for nonsimple closed curves. Proc. Am. Math. Soc. 118, 197–203 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hadwiger, H.: Die isoperimetrische Ungleichung in Raum. Elemente Math. 3, 25–38 (1948)

    MATH  MathSciNet  Google Scholar 

  15. Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin (1957)

    Book  MATH  Google Scholar 

  16. Kotlyar, B.D.: On a geometric inequality. (UDC 513:519.21) Ukrainskii Geometricheskii Sbornik. 30, 49–52 (1987)

    Google Scholar 

  17. Ku, H., Ku, M., Zhang, X.: Isoperimetric inequalities on surfaces of constant curvature. Can. J. Math. 49, 1162–1187 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  20. Osserman, R.: Bonnesen-style isoperimetric inequality. Am. Math. Mon. 86, 1–29 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pleijel, A.: On konvexa kurvor. Nordisk Math. Tidskr. 3, 57–64 (1955)

    MATH  MathSciNet  Google Scholar 

  22. Polya, G., Szego, G.: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, vol. 27. Princeton University Press, Princeton (1951)

    Google Scholar 

  23. Ren, D.: Topics in Integral Geometry. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  24. Sangwine-Yager, J.R.: Mixe Volumes. In: Gruber P., Wills, J. (eds.) Handbook of Convex Geometry, vol. A, pp. 43–71. North-Holland, Amsterdam (1993)

    Chapter  Google Scholar 

  25. Santaló, L.A.: Integral Geometry and Geometric Probability. Addison-Wesley, Reading (1976)

    MATH  Google Scholar 

  26. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  27. Stone, A.: On the isoperimetric inequality on a minimal surface. Calc. Var. Partial Differ. Equ. 17(4), 369–391 (2003)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Wei, S., Zhu, M.: Sharp isoperimetric inequalities and sphere theorems. Pac. J. Math. 220, 183–195 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Weiner, J.L.: A generalization of the isoperimetric inequality on the 2-sphere. Indiana Univ. Math. J. 24, 243–248 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  32. Weiner, J.L.: Isoperimetric inequalities for immersed closed spherical curves. Proc. Am. Math. Soc. 120, 501–506 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Xu, W., Zhou, J., Zhu, B.: On containment measure and the mixed isoperimetric inequality. J. Inequal. Appl. 2013, 540 (2013)

    Google Scholar 

  34. Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Ec. Norm. Super. Paris 8, 487–507 (1975)

    MATH  Google Scholar 

  35. Zeng, C., Ma, L., Zhou, J.: The Bonnesen isoperimetric inequality in a surface of constant curvature. Sci. China Math. 55, 1913–1919 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zeng, C., Zhou, J., Yue, S.: The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math. Sin. 55, 355–362 (2012)

    MATH  MathSciNet  Google Scholar 

  37. Zhang, X-M.: Schur-convex functions and isoperimetric inequalities. Proc. Am. Math. Soc. 126, 461–470 (1998)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  39. Zhao, L., Ma, L., Zhou, J.: A geometric application of the Wirtinger inequality. J. Math. 31, 887–890 (2011)

    MATH  MathSciNet  Google Scholar 

  40. Zhou, J.: On Bonnesen-type inequalities. Acta. Math. Sin. Chin. Ser. 50, 1397–1402 (2007)

    MATH  Google Scholar 

  41. Zhou, J., Chen, F.: The Bonnesen-type inequalities in a plane of constant curvature. J. Kor. Math. Soc. 44, 1–10 (2007)

    Article  Google Scholar 

  42. Zhou, J., Du, Y., Cheng, F.: Some Bonnesen-style inequalities for higher dimensions. Acta Math. Sin. (English Series) 28, 2561–2568 (2012)

    Google Scholar 

  43. Zhou, J., Ma, L., Xu, W.: On the isoperimetric deficit upper Limit. Bull. Kor. Math. Soc. 50, 175–184 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  44. Zhou, J., Ren, D.: Geometric inequalities - from integral geometry point of view. Acta Math. Scientia Ser. A. 30, 1322–1339 (2010)

    MATH  MathSciNet  Google Scholar 

  45. Zhou, J., Xia, Y. Zeng, C.: Some new Bonnesen-style inequalities. J. Kor. Math. Soc. 48, 421–430 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. Zhou, J., Zhou, C., Ma, F.: Isoperimetric deficit upper limit of a planar convex set. Rendiconti del Circolo Matematico di Palermo (Serie II, Suppl), 81, 363–367 (2009)

    Google Scholar 

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Acknowledgements

We would like to thank the anonymous referee for helpful comments and suggestions. This research is supported in part by NSFC (grant no. 11271302) and the Ph.D. Program of Higher Education Research Fund (grant no. 2012182110020).

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Authors and Affiliations

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

    Wenxue Xu, Jiazu Zhou & Baocheng Zhu

  2. College of Mathematical Science, Kaili University, Kaili, Guizhou, 556000, China

    Jiazu Zhou

  3. Department of Mathematics, Hubei University for Nationalities, Enshi, Huibei, 445000, China

    Baocheng Zhu

Authors
  1. Wenxue Xu
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  2. Jiazu Zhou
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Corresponding author

Correspondence to Jiazu Zhou .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea

    Young Jin Suh

  2. Department of Mathematics, King’s College London, London, UK

    Jürgen Berndt

  3. Department of Mathematics, Osaka City University, Osaka, Japan

    Yoshihiro Ohnita

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Republic of Korea

    Byung Hak Kim

  5. The Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang, Republic of Korea

    Hyunjin Lee

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Xu, W., Zhou, J., Zhu, B. (2014). Bonnesen-Style Symmetric Mixed Isoperimetric Inequality. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_9

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