대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제50권1호
  • /
  • Pages.175-184
  • /
  • 2013
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT

  • Zhou, Jiazu (School of Mathematics and Statistics Southwest University, Southeast Guizhou Vocational College of Technology for Nationalities) ;
  • Ma, Lei (Southeast Guizhou Vocational College of Technology for Nationalities) ;
  • Xu, Wenxue (School of Mathematics and Statistics Southwest University, Southeast Guizhou Vocational College of Technology for Nationalities)
  • 투고 : 2011.07.10
  • 발행 : 2013.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, the reverse Bonnesen style inequalities for convex domain in the Euclidean plane $\mathbb{R}^2$ are investigated. The Minkowski mixed convex set of two convex sets K and L is studied and some new geometric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style inequalities for convex domain K are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema ([5]) and Pleijel ([22]).

키워드

참고문헌

  1. T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175-192.
  2. J. Bokowski and E. Heil, Integral representation of quermassintegrals and Bonnesenstyle inequalities, Arch. Math. (Basel) 47 (1986), no. 1, 79-89. https://doi.org/10.1007/BF01202503
  3. T. Bonnesen, Les problems des isoperimetres et des isepiphanes, Gauthier-Villars, Paris, 1929.
  4. T. Bonnesen and W. Fenchel, Theorie der konvexen Koeper, 2nd ed., Berlin-Heidelberg-New York, 1974.
  5. O. Bottema, Eine obere Grenze fur das isoperimetrische Defizit ebener Kurven, Nederl. Akad. Wetensch. Proc. A66 (1933), 442-446.
  6. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag Berlin Heidelberg, 1988.
  7. V. Diskant, A generalization of Bonnesen's inequalities, Soviet Math. Dokl. 14 (1973), 1728-1731 (Transl. of Dokl. Akad. Nauk SSSR 213 (1973), 519-521).
  8. H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), 581-593. https://doi.org/10.2307/2313773
  9. E. Grinberg, S. Li, G. Zhang, and J. Zhou, Integral Geometry and Convexity, Proceedings of the International Conference, World Scientific, 2006.
  10. E. Grinberg, D. Ren, and J. Zhou, The symetric isoperimetric deficit and the containment problem in a plan of constant curvature, preprint.
  11. L. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197-203. https://doi.org/10.1090/S0002-9939-1993-1079698-X
  12. H. Hadwiger, Die isoperimetrische Ungleichung in Raum, Elemente der Math. 3 (1948), 25-38.
  13. H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer, Berlin, 1957.
  14. G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambradge Univ. Press, Cambradge/New York, 1952.
  15. R. Howard, The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2779-2787. https://doi.org/10.1090/S0002-9939-98-04336-6
  16. W. Y. Hsiang, An elementary proof of the isoperimetric problem, Chinese Ann. Math. Ser. A 23 (2002), no. 1, 7-12.
  17. C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. 73 (1961), no. 2, 213-220. https://doi.org/10.2307/1970287
  18. H. Ku, M. Ku, and X. Zhang, Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math. 49 (1997), no. 6, 1162-1187. https://doi.org/10.4153/CJM-1997-057-x
  19. M. Li and J. Zhou, An upper limit for the isoperimetric deficit of convex set in a plane of constant curvature, Sci. in China 53 (2010), no. 8, 1941-1946. https://doi.org/10.1007/s11425-010-4018-3
  20. R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182-1238. https://doi.org/10.1090/S0002-9904-1978-14553-4
  21. R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly 86 (1979), no. 1, 1-29. https://doi.org/10.2307/2320297
  22. A. Pleijel, On konvexa kurvor, Nordisk Math. Tidskr. 3 (1955), 57-64.
  23. G. Polya and G. Szego, Isoperimetric inequalities in mathematical physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
  24. D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
  25. L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison-Wesley, 1976.
  26. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.
  27. E. Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. 57 (1991), no. 5, 508-513. https://doi.org/10.1007/BF01246751
  28. E. Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622-630. https://doi.org/10.1007/BF03323109
  29. J. L. Weiner, A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974), 243-248. https://doi.org/10.1512/iumj.1974.24.24021
  30. J. L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501-506. https://doi.org/10.1090/S0002-9939-1994-1163337-4
  31. S. T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. Ec. Norm. Super. Paris 8 (1975), no. 4, 487-507. https://doi.org/10.24033/asens.1299
  32. G. Zhang and J. Zhou, Containment measures in integral geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006.
  33. J. Zhou, On Bonnesen-type inequalities, Acta. Math. Sinica, Chinese Series 50 (2007), no. 6, 1397-1402.
  34. J. Zhou and F. Chen, The Bonnesen-type inequalities in a plane of constant curvature, J. Korean Math. Soc. 44 (2007), no. 6, 1363-1372. https://doi.org/10.4134/JKMS.2007.44.6.1363
  35. J. Zhou, Y. Du, and F. Cheng, Some Bonnesen-style inequalities for higher dimensions, to appear in Acta. Math. Sinica.
  36. J. Zhou and L. Ma, The discrete isoperimetric deficit upper bound, preprint.
  37. J. Zhou and D. Ren, Geometric inequalities from the viewpoint of integral geometry, Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 5, 1322-1339.
  38. J. Zhou, Y. Xia, and C. Zeng, Some new Bonnesen-style inequalities, J. Korean Math. Soc. 48 (2011), no. 2, 421-430. https://doi.org/10.4134/JKMS.2011.48.2.421
  39. C. Zeng, J. Zhou, and S. Yue, The symmetric mixed isoperimetric inequality of two planar convex domains, Acta Math. Sinica 55 (2012), no. 3, 355-362.

피인용 문헌

  1. Bonnesen-style Wulff isoperimetric inequality vol.2017, pp.1, 2017, https://doi.org/10.1186/s13660-017-1305-3
  2. On containment measure and the mixed isoperimetric inequality vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-540
  3. Reverse Bonnesen style inequalities in a surface $$\mathbb{X}_\varepsilon ^2$$ of constant curvature vol.56, pp.6, 2013, https://doi.org/10.1007/s11425-013-4578-0
  4. Bonnesen-style symmetric mixed inequalities vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1146-5