Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME NEW BONNESEN-STYLE INEQUALITIES

  • Zhou, Jiazu (SCHOOL OF MATHEMATICS AND STATISTICS SOUTHWEST UNIVERSITY) ;
  • Xia, Yunwei (SCHOOL OF MATHEMATICS AND STATISTICS SOUTHWEST UNIVERSITY) ;
  • Zeng, Chunna (SCHOOL OF MATHEMATICS AND STATISTICS SOUTHWEST UNIVERSITY)
  • Received : 2009.11.03
  • Published : 2011.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

By evaluating the containment measure of one domain to contain another, we will derive some new Bonnesen-type inequalities (Theorem 2) via the method of integral geometry. We obtain Ren's sufficient condition for one domain to contain another domain (Theorem 4). We also obtain some new geometric inequalities. Finally we give a simplified proof of the Bottema's result.

Keywords

References

  1. T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175-192. https://doi.org/10.4310/jdg/1214430403
  2. J. Bokowski and E. Heil, Integral representations of quermassintegrals and Bonnesen- style 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, Paris, 1929.
  4. T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Springer-Verlag, Berlin- New York, 1974.
  5. O. Bottema, Eine obere Grenze fur das isoperimetrische Dezit 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. C. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), no. 2, 187-192. https://doi.org/10.1007/BF02566344
  8. V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Akad. Nauk SSSR 213 (1973), 519-521.
  9. K. Enomoto, A generalization of the isoperimetric inequality on $S^{2}$ and flat tori in $S^{3}$, Proc. Amer. Math. Soc. 120 (1994), no. 2, 553-558.
  10. E. Grinberg, D. Ren, and J. Zhou, The symetric isoperimetric deficit and the contain- ment problem in a plan of constant curvature, preprint.
  11. E. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies, Math. Ann. 291 (1991), no. 1, 75-86. https://doi.org/10.1007/BF01445191
  12. E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77-115. https://doi.org/10.1112/S0024611599001653
  13. 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
  14. G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambradge Univ. Press, Cambradge/New York, 1951.
  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. C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220. https://doi.org/10.2307/1970287
  17. W. Y. Hsiung, An elementary proof of the isoperimetric problem, Chinese Ann. Math. Ser. A 23 (2002), no. 1, 7-12.
  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 decit of convex set in a plane of constant curvature, Sci. in China Mathematics 53 (2010), no. 8, 1941-1946. https://doi.org/10.1007/s11425-010-4018-3
  20. E. Lutwak, D. Yang, and G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38. https://doi.org/10.4310/jdg/1090425527
  21. 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
  22. R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1-29. https://doi.org/10.2307/2320297
  23. A. Pleijel, On konvexa kurvor, Nordisk Math. Tidskr. 3 (1955), 57-64.
  24. G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
  25. D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
  26. L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
  27. A. Stone, On the isoperimetric inequality on a minimal surface, Calc. Var. Partial Differential Equations 17 (2003), no. 4, 369-391. https://doi.org/10.1007/s00526-002-0174-9
  28. D. Tang, Discrete Wirtinger and isoperimetric type inequalities, Bull. Austral. Math. Soc. 43 (1991), no. 3, 467-474. https://doi.org/10.1017/S0004972700029312
  29. E. Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508-513. https://doi.org/10.1007/BF01246751
  30. 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
  31. S. Wei and M. Zhu, Sharp isoperimetric inequalities and sphere theorems, Pacific J. Math. 220 (2005), no. 1, 183-195. https://doi.org/10.2140/pjm.2005.220.183
  32. J. L. Weiner, A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974/75), 243-248. https://doi.org/10.1512/iumj.1974.24.24021
  33. 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
  34. S. T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. Ecole Norm. Sup. (4) 8 (1975), no. 4, 487-507. https://doi.org/10.24033/asens.1299
  35. G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202. https://doi.org/10.4310/jdg/1214425451
  36. G. Zhang and J. Zhou, Containment measures in integral geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006.
  37. X.-M. Zhang, Bonnesen-style inequalities and pseudo-perimeters for polygons, J. Geom. 60 (1997), no. 1-2, 188-201. https://doi.org/10.1007/BF01252226
  38. X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470. https://doi.org/10.1090/S0002-9939-98-04151-3
  39. J. Zhou, A kinematic formula and analogues of Hadwiger's theorem in space, Geometric analysis (Philadelphia, PA, 1991), 159-167, Contemp. Math., 140, Amer. Math. Soc., Providence, RI, 1992.
  40. J. Zhou, The sufficient condition for a convex body to enclose another in TEX>$R^{4}$, Proc. Amer. Math. Soc. 121 (1994), no. 3, 907-913.
  41. J. Zhou, Kinematic formulas for mean curvature powers of hypersurfaces and Had- wiger's theorem in $R^{2n}$, Trans. Amer. Math. Soc. 345 (1994), no. 1, 243-262. https://doi.org/10.2307/2154603
  42. J. Zhou, When can one domain enclose another in $R^{3}$?, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 266-272. https://doi.org/10.1017/S1446788700038660
  43. J. Zhou, Sufficient conditions for one domain to contain another in a space of constant curvature, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2797-2803. https://doi.org/10.1090/S0002-9939-98-04369-X
  44. J. Zhou, On Willmore's inequality for submanifolds, Canad. Math. Bull. 50 (2007), no. 3, 474-480. https://doi.org/10.4153/CMB-2007-047-4
  45. J. Zhou, On the Willmore deficit of convex surfaces, Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993), 279-287, Lectures in Appl. Math., 30, Amer. Math. Soc., Providence, RI, 1994.
  46. J. Zhou, The Willmore functional and the containment problem in $R^{4}$, Sci. China Ser. A 50 (2007), no. 3, 325-333. https://doi.org/10.1007/s11425-007-0029-0
  47. J. Zhou, Bonnesen-type inequalities on the plane, Acta Math. Sinica (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
  48. 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

Cited by

  1. The Bonnesen isoperimetric inequality in a surface of constant curvature vol.55, pp.9, 2012, https://doi.org/10.1007/s11425-012-4405-z
  2. 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
  3. Some Bonnesen-style inequalities for higher dimensions vol.28, pp.12, 2012, https://doi.org/10.1007/s10114-012-9657-6
  4. ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT vol.50, pp.1, 2013, https://doi.org/10.4134/BKMS.2013.50.1.175
  5. On containment measure and the mixed isoperimetric inequality vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-540
  6. Bonnesen-style symmetric mixed inequalities vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1146-5
  7. Bonnesen-style inequalities on surfaces of constant curvature vol.2018, pp.1, 2018, https://doi.org/10.1186/s13660-018-1918-1
  8. Reverse Bonnesen-style inequalities on surfaces of constant curvature vol.29, pp.06, 2018, https://doi.org/10.1142/S0129167X18500404