The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains

Chun Na ZENG, Jia Zu ZHOU, Shuang Shan YUE

Acta Mathematica Sinica, Chinese Series ›› 2012 ›› Issue (2) : 355-362.

PDF(467 KB)
中国科学院数学与系统科学研究院期刊网
PDF(467 KB)
Acta Mathematica Sinica, Chinese Series ›› 2012 ›› Issue (2) : 355-362. DOI: 10.12386/A2012sxxb0035
Articles

The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains

  • Chun Na ZENG1, Jia Zu ZHOU1,2, Shuang Shan YUE1
Author information +
History +

Abstract

Let Kk(k=#em/em#,j) be domain of the area Ak, and of the perimeter Pk, respectively. The symmetric mixed isoperimetric deficit of K#em/em# and Kj is defined as σ(K#em/em#,Kj)=P#em/em#2Pj2-16π2A#em/em#Aj. We follow the ideas of Zhou, Ren (2010) and Zhou, Yue (2009) and obtain some Bonnesen-style symmetric mixed inequalities and the symmetric mixed isoperimetric inequality by the method of integral geometry. We also obtain some symmetric mixed isoperimetric upper limits. Some discrete Bonnesen-style symmetric mixed inequalities and one upper limit of the discrete symmetric mixed isoperimetric deficit for two domains are obtained. Finally we apply these symmetric mixed (isoperimetric) inequalities to estimate the complete elliptic integral of second class.

Key words

The symmetric mixed isoperimetric deficit / the symmetric mixed isoperimetric inequality / the Bonnesen-style symmetric mixed inequality

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations
Chun Na ZENG, Jia Zu ZHOU, Shuang Shan YUE. The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains. Acta Mathematica Sinica, Chinese Series, 2012(2): 355-362 https://doi.org/10.12386/A2012sxxb0035

References

[1] Zhou J., Ren D., Geometric inequalities-from integral geometry point of view, Acta Mathematica Scientia, 2010, 30A(5): 1322-1339.

[2] Zhou J. Z., Yue J., On the isohomothetic inequalities, 2009, preprint.

[3] Bonnesen T., Les Probl閙es des Isop閞im閠res at des Is閜iphanes, Paris: Gauthie-Villars, 1920.

[4] Bonnesen T., Uber eine Verschärfung der isoperimetrischen Ungleichheit des Kreises in der Ebene und auf der Kugeloberfláche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper, Math. Annalen, 1921, 84: 216-227.

[5] Bonnesen T., Uber das isoperimetrische Defizit ebener Figuren, Math. Annalen, 1924, 91: 252-268.

[6] Bonnesen T., Quelques probl閙s isop閞imetriques, Acta Mathematica, 1926, 48: 123-178.

[7] Burago Yu D., Zalgaller V. A., Geometric Inequalities, Berlin, Heidelberg: Springer-Verlag, 1998.

[8] Osserman R., The isoperimetric inequality, Bull. Amer. Math. Soc., 1978, 84: 1182-1238.

[9] Osserman R., Bonnesen-style isoperimetric inequality, Amer. Math. Monthly, 1979, 86: 1-29.

[10] Banchoff T. F., Pohl W. F., Ageneralization of the isoperimetric inequality, J. Diff. Geo., 1971, 6: 175-213.

[11] Bokowski J., Heil E., Integral representation of quermassintegrals and Bonnesen-style inequalities, Arch. Math., 1986, 47: 79-89.

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

[13] Diskant V., A generalization of Bonnesen's inequalities, Soviet Math. Dokl., 1973, 14: 1728-1731 (Transl. of Dokl. Akad. Nauk SSSR, 1973, 213).

[14] Kotlyar B. D., On a Geometricheskí inequality, Ukrainskí Geometricheskí Sbornik, 1987, 30: 49-52.

[15] Grinberg E., Ren D., Zhou J., The symmetric isoperimetric deficit and the containment problem in a plan of constant curvature, 1994, preprint.

[16] Grinberg E., Li S., Zhang G., Zhou J., Integral geometry and Convexity, Sigapore: World Scientific, 2006.

[17] Grinberg E., Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies, Math. Ann., 1991, 291: 75-86.

[18] Enomoto K., A generalization of the isoperimetric inequality on S2 and flat tori in S3, Proc. Amer. Math. Soc., 1994, 120(2): 553-558.

[19] Ren D., Topics in Integral Geometry, Sigapore: World Scientific, 1994.

[20] Santaló L. A., Integral Geometry and Geometric Probability, Addison-Wesley, Reading, Mass., 1976.

[21] Gysin L., The isoperimetric inequality for nonsimple closed curves, Proc. Amer. Math. Soc., 1993, 118(1): 197-203.

[22] Hadwiger H., Die isoperimetrische Ungleichung in Raum, Elemente Math., 1948, 3: 25-38.

[23] Howard R., The sharp sobolev inequality and the Banchoff-Pohl inequality on surfaces, Proc. Amer. Math. Soc., 1998, 126: 2779-2787.

[24] Hsiung C. C., Isoperimetric inequalities for two-dimensinal Riemannian manifolds with boundary, Ann. of Math., 1961, 73(2): 213-220.

[25] Hsiung W. Y., An elementary proof of the isoperimetric problem, Chin. Ann. Math., 2002, 23A(1): 7-12.

[26] Ku H., Ku M., Zhang X., Isoperimetric inequalities on surfaces of constant curvature, Canadian J. of Math., 1997, 49: 1162-1187.

[27] Zhou J. Z., On Bonnesen-type inequalities, Acta Mathematica Sinica, Chinese Series, 2007, 50(6): 1397-1402.

[28] Zhou J., Ma L., Upper bound of the isoperimetric defict, preprint.

[29] Li M., Zhou J., An upper limit for the isoperimetric deficit of convex set in a plane of constant curvature, Sci. in China, 2010, 53(8): 1941-1946.

[30] Zhou J., Xia Y., Zeng C., Some new Bonnesen-style inequalities, J. Korean Math. Soc., 2011, 48(2): 421-430.

[31] Zhou J., Chen F., The Bonnesen-type inequalities in a plane of constant curvature, Journal of Korean Math. Sco., 2007, 44(6): 1363-1372.

[32] Klain D., Bonnesen-type inequalities for surfaces of constant curvature, Adv. in Appl. Math., 2007, 39(2): 143-154.

[33] Zhou J., Integral geometry and the isoperimetric inequality, J. Guizhou Normal Univ. (Natural Science), 2002, 20(2): 1-5.

[34] Zhou J., A kinematic formula and analogous of Hadwiger's theorem in space, Contemporary Mathematics, 1992, 140: 159-167.

[35] Schneide R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge: Cambridge University Press, 1993.

[36] Ren D., Introduction to Integral Geometry, Shanghai: Scientific Technical Publishers, 1998.

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

[38] Bottema O., Eine obere Grenze für das isoperimetrische Defizit ebener Kurven, Nederl. Akad. Wetensch. Proc., 1933, A66: 442-446.

[39] Pleijel A., On konvexa kurvor, Nordisk Math. Tidskr., 1955, 3: 57-64.

[40] Berger M., Geometry I, Berlin: Springer-Verlag, World Publishing Corporation, 1989.

[41] Zhou Z., The approximative formula for circum length of an ellipse, Shuxue Tongbao, 1996, 6: 45-47.
PDF(467 KB)

243

Accesses

0

Citation

Detail
Sections
Recommended

/

〈 〉