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http://dx.doi.org/10.4134/JKMS.2011.48.2.421

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)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.2, 2011 , pp. 421-430 More about this Journal
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
the isoperimetric inequality; kinematic formula; containment measure; convex domain; the Bonnesen-style inequality;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
Times Cited By Web Of Science : 2  (Related Records In Web of Science)
Times Cited By SCOPUS : 3
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1 J. Zhou, When can one domain enclose another in $R^{3}$?, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 266-272.   DOI
2 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.   DOI   ScienceOn
3 J. Zhou, On Willmore's inequality for submanifolds, Canad. Math. Bull. 50 (2007), no. 3, 474-480.   DOI
4 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.
5 J. Zhou, The Willmore functional and the containment problem in $R^{4}$, Sci. China Ser. A 50 (2007), no. 3, 325-333.   DOI   ScienceOn
6 J. Zhou, Bonnesen-type inequalities on the plane, Acta Math. Sinica (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
7 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.   과학기술학회마을   DOI   ScienceOn
8 E. Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622-630.   DOI
9 S. Wei and M. Zhu, Sharp isoperimetric inequalities and sphere theorems, Pacific J. Math. 220 (2005), no. 1, 183-195.   DOI
10 J. L. Weiner, A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974/75), 243-248.   DOI
11 J. L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501-506.   DOI   ScienceOn
12 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.   DOI
13 G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202.   DOI
14 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.
15 G. Zhang and J. Zhou, Containment measures in integral geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006.
16 X.-M. Zhang, Bonnesen-style inequalities and pseudo-perimeters for polygons, J. Geom. 60 (1997), no. 1-2, 188-201.   DOI
17 X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470.   DOI   ScienceOn
18 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.
19 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.   DOI   ScienceOn
20 W. Y. Hsiung, An elementary proof of the isoperimetric problem, Chinese Ann. Math. Ser. A 23 (2002), no. 1, 7-12.
21 H. Ku, M. Ku, and X. Zhang, Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math. 49 (1997), no. 6, 1162-1187.   DOI   ScienceOn
22 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.   DOI
23 E. Lutwak, D. Yang, and G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38.   DOI
24 R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182-1238.   DOI
25 D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
26 R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1-29.   DOI   ScienceOn
27 A. Pleijel, On konvexa kurvor, Nordisk Math. Tidskr. 3 (1955), 57-64.
28 G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
29 L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
30 A. Stone, On the isoperimetric inequality on a minimal surface, Calc. Var. Partial Differential Equations 17 (2003), no. 4, 369-391.   DOI
31 D. Tang, Discrete Wirtinger and isoperimetric type inequalities, Bull. Austral. Math. Soc. 43 (1991), no. 3, 467-474.   DOI
32 E. Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508-513.   DOI   ScienceOn
33 T. Bonnesen and W. Fenchel, Theorie der konvexen Korper, Springer-Verlag, Berlin- New York, 1974.
34 O. Bottema, Eine obere Grenze fur das isoperimetrische Dezit ebener Kurven, Nederl. Akad. Wetensch. Proc. A66 (1933), 442-446.
35 Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, Heidelberg, 1988.
36 C. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), no. 2, 187-192.   DOI
37 V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Akad. Nauk SSSR 213 (1973), 519-521.
38 E. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies, Math. Ann. 291 (1991), no. 1, 75-86.   DOI
39 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.
40 E. Grinberg, D. Ren, and J. Zhou, The symetric isoperimetric deficit and the contain- ment problem in a plan of constant curvature, preprint.
41 E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77-115.   DOI
42 L. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197-203.   DOI   ScienceOn
43 G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambradge Univ. Press, Cambradge/New York, 1951.
44 R. Howard, The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2779-2787.   DOI   ScienceOn
45 C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220.   DOI
46 T. Bonnesen, Les problems des isoperimetres et des isepiphanes, Paris, 1929.
47 T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175-192.   DOI
48 J. Bokowski and E. Heil, Integral representations of quermassintegrals and Bonnesen- style inequalities, Arch. Math. (Basel) 47 (1986), no. 1, 79-89.   DOI   ScienceOn