On Bonnesen-type inequalities for a surface of constant curvature

Xu, Wenxue; Zhou, Jiazu; Zhu, Baocheng

doi:10.1090/proc/12657

On Bonnesen-type inequalities for a surface of constant curvature
HTML articles powered by AMS MathViewer

by Wenxue Xu, Jiazu Zhou and Baocheng Zhu PDF

Proc. Amer. Math. Soc. 143 (2015), 4925-4935 Request permission

Abstract:

New Bonnesen-type inequalities for simply connected domains on surfaces of constant curvature are proved by using integral formulas. These inequalities are generalizations of known inequalities of convex domains.

References

Thomas F. Banchoff and William F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175–192. MR 305319
Jürgen Bokowski and Erhard Heil, Integral representations of quermassintegrals and Bonnesen-style inequalities, Arch. Math. (Basel) 47 (1986), no. 1, 79–89. MR 855141, DOI 10.1007/BF01202503
T. Bonnesen, Les probléms des isopérimétres et des isépiphanes, Paris, 1929.
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
Stefano Campi, Three-dimensional Bonnesen type inequalities, Matematiche (Catania) 60 (2005), no. 2, 425–431 (2006). MR 2257044
V. I. Diskant, A generalization of Bonnesen’s inequalities, Dokl. Akad. Nauk SSSR 213 (1973), 519–521 (Russian). MR 0338925
Kazuyuki 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. MR 1163333, DOI 10.1090/S0002-9939-1994-1163333-7
Mark Green and Stanley Osher, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asian J. Math. 3 (1999), no. 3, 659–676. MR 1793675, DOI 10.4310/AJM.1999.v3.n3.a5
E. Grinberg, D. Ren, and J. Zhou, The symmetric isoperimetric deficit and the containment problem in a plan of constant curvature, preprint.
Eric L. Grinberg, Shougui Li, Gaoyong Zhang, and Jiazu Zhou (eds.), Integral geometry and convexity, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2246246, DOI 10.1142/5950
H. Hadwiger, Die isoperimetrische Ungleichung im Raum, Elem. Math. 3 (1948), 25–38 (German). MR 24641
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775
Chuan-chih Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213–220. MR 130637, DOI 10.2307/1970287
Daniel A. Klain, Bonnesen-type inequalities for surfaces of constant curvature, Adv. in Appl. Math. 39 (2007), no. 2, 143–154. MR 2333645, DOI 10.1016/j.aam.2006.11.004
Daniel A. Klain, An error estimate for the isoperimetric deficit, Illinois J. Math. 49 (2005), no. 3, 981–992. MR 2210271
Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
Hsu-Tung Ku, Mei-Chin Ku, and Xin-Min Zhang, Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math. 49 (1997), no. 6, 1162–1187. MR 1611644, DOI 10.4153/CJM-1997-057-x
Ming Li and JiaZu Zhou, An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature, Sci. China Math. 53 (2010), no. 8, 1941–1946. MR 2679076, DOI 10.1007/s11425-010-4018-3
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
L. A. Santaló, Integral geometry on surfaces of constant negative curvature, Duke Math. J. 10 (1943), 687–709. MR 9469
L. A. Santaló, Integral formulas in Crofton’s style on the sphere and some inequalities referring to spherical curves, Duke Math. J. 9 (1942), 707–722. MR 8160
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Dingyi Tang, Discrete Wirtinger and isoperimetric type inequalities, Bull. Austral. Math. Soc. 43 (1991), no. 3, 467–474. MR 1107401, DOI 10.1017/S0004972700029312
Eberhard Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508–513. MR 1129528, DOI 10.1007/BF01246751
Eberhard Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622–630. MR 1174928, DOI 10.1007/BF03323109
Shihshu Walter Wei and Meijun Zhu, Sharp isoperimetric inequalities and sphere theorems, Pacific J. Math. 220 (2005), no. 1, 183–195. MR 2195069, DOI 10.2140/pjm.2005.220.183
Joel L. Weiner, A generalization of the isoperimetric inequality on the $2$-sphere, Indiana Univ. Math. J. 24 (1974/75), 243–248. MR 380687, DOI 10.1512/iumj.1974.24.24021
YunWei Xia, WenXue Xu, JiaZu Zhou, and BaoCheng Zhu, Reverse Bonnesen style inequalities in a surface $\Bbb {X}_\varepsilon ^2$ of constant curvature, Sci. China Math. 56 (2013), no. 6, 1145–1154. MR 3063961, DOI 10.1007/s11425-013-4578-0
Wenxue Xu, Jiazu Zhou, and Baocheng Zhu, On containment measure and the mixed isoperimetric inequality, J. Inequal. Appl. , posted on (2013), 2013:540, 11. MR 3212973, DOI 10.1186/1029-242X-2013-540
ChunNa Zeng, Lei Ma, JiaZu Zhou, and FangWei Chen, The Bonnesen isoperimetric inequality in a surface of constant curvature, Sci. China Math. 55 (2012), no. 9, 1913–1919. MR 2960869, DOI 10.1007/s11425-012-4405-z
Chun Na Zeng, Jia Zu Zhou, and Shuang Shan Yue, A symmetric mixed isoperimetric inequality for two planar convex domains, Acta Math. Sinica (Chinese Ser.) 55 (2012), no. 2, 355–362 (Chinese, with English and Chinese summaries). MR 2963004
Gao Yong Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95–116. MR 1288055, DOI 10.1112/S0025579300007208
Gaoyong Zhang and Jiazu Zhou, Containment measures in integral geometry, Integral geometry and convexity, World Sci. Publ., Hackensack, NJ, 2006, pp. 153–168. MR 2240979, DOI 10.1142/9789812774644_{0}011
Xin-Min Zhang, Bonnesen-style inequalities and pseudo-perimeters for polygons, J. Geom. 60 (1997), no. 1-2, 188–201. MR 1477080, DOI 10.1007/BF01252226
Xin-Min Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc. 126 (1998), no. 2, 461–470. MR 1423343, DOI 10.1090/S0002-9939-98-04151-3
Jia Zu Zhou, Bonnesen-type inequalities on the plane, Acta Math. Sinica (Chinese Ser.) 50 (2007), no. 6, 1397–1402 (Chinese, with English and Chinese summaries). MR 2516365
Jiazu Zhou and Fangwei Chen, The Bonnesen-type inequalities in a plane of constant curvature, J. Korean Math. Soc. 44 (2007), no. 6, 1363–1372. MR 2358960, DOI 10.4134/JKMS.2007.44.6.1363
Jia Zu Zhou, Yan Hua Du, and Fei Cheng, Some Bonnesen-style inequalities for higher dimensions, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 12, 2561–2568. MR 2996526, DOI 10.1007/s10114-012-9657-6
Jiazu Zhou, Lei Ma, and Wenxue Xu, On the isoperimetric deficit upper limit, Bull. Korean Math. Soc. 50 (2013), no. 1, 175–184. MR 3029540, DOI 10.4134/BKMS.2013.50.1.175
Jia Zu Zhou and De Lin Ren, Geometric inequalities from the viewpoint of integral geometry, Acta Math. Sci. Ser. A (Chinese Ed.) 30 (2010), no. 5, 1322–1339 (Chinese, with English and Chinese summaries). MR 2780155
Jiazu Zhou, Yunwei Xia, and Chunna Zeng, Some new Bonnesen-style inequalities, J. Korean Math. Soc. 48 (2011), no. 2, 421–430. MR 2789464, DOI 10.4134/JKMS.2011.48.2.421