SOME NEW BONNESEN-STYLE INEQUALITIES

Title & Authors
SOME NEW BONNESEN-STYLE INEQUALITIES
Zhou, Jiazu; Xia, Yunwei; Zeng, Chunna;

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;
Language
English
Cited by
1.
ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT,;;;

대한수학회보, 2013. vol.50. 1, pp.175-184
1.
The Bonnesen isoperimetric inequality in a surface of constant curvature, Science China Mathematics, 2012, 55, 9, 1913
2.
Reverse Bonnesen style inequalities in a surface $$\mathbb{X}_\varepsilon ^2$$ of constant curvature, Science China Mathematics, 2013, 56, 6, 1145
3.
Some Bonnesen-style inequalities for higher dimensions, Acta Mathematica Sinica, English Series, 2012, 28, 12, 2561
4.
ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT, Bulletin of the Korean Mathematical Society, 2013, 50, 1, 175
5.
On containment measure and the mixed isoperimetric inequality, Journal of Inequalities and Applications, 2013, 2013, 1, 540
6.
Bonnesen-style symmetric mixed inequalities, Journal of Inequalities and Applications, 2016, 2016, 1
References
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 representations of quermassintegrals and Bonnesen- style inequalities, Arch. Math. (Basel) 47 (1986), no. 1, 79-89.

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.

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.

12.
E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77-115.

13.
L. Gysin, The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc. 118 (1993), no. 1, 197-203.

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.

16.
C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220.

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.

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.

20.
E. Lutwak, D. Yang, and G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38.

21.
R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182-1238.

22.
R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1-29.

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.

28.
D. Tang, Discrete Wirtinger and isoperimetric type inequalities, Bull. Austral. Math. Soc. 43 (1991), no. 3, 467-474.

29.
E. Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508-513.

30.
E. Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622-630.

31.
S. Wei and M. Zhu, Sharp isoperimetric inequalities and sphere theorems, Pacific J. Math. 220 (2005), no. 1, 183-195.

32.
J. L. Weiner, A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974/75), 243-248.

33.
J. L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501-506.

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.

35.
G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202.

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.

38.
X.-M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461-470.

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.

42.
J. Zhou, When can one domain enclose another in $R^{3}$?, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 266-272.

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.

44.
J. Zhou, On Willmore's inequality for submanifolds, Canad. Math. Bull. 50 (2007), no. 3, 474-480.

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.

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.