References
- T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geom. 6 (1971), 175-213. https://doi.org/10.4310/jdg/1214430403
- W. Blaschke, Vorlesungen uber Intergralgeometrie, 3rd ed. Deutsch. VerlagWiss., Berlin, 1955.
- J. Bokowski and E. Heil, Integral representation of quermassintegrals and Bonnesenstyle inequalities, Arch. Math. 47 (1986), no. 1, 79-89. https://doi.org/10.1007/BF01202503
- T. Bonnesen, Les problems des isoperimetres et des isepiphanes, Paris, 1929.
- T. Bonnesen and W. Fenchel, Theorie der konvexen Koeper, 2nd ed., Berlin-Heidelberg-New York, 1974.
- Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag Berlin Heidelberg, 1988.
- A. Dinghas, Bemerkung zu einer Verscharfung der isoperimetrischen Ungleichung durch H. Hadwiger, Math. Nachr. 1 (1948), 284-286. https://doi.org/10.1002/mana.19480010503
- V. Diskant, Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference, Sibirskii Mat. Zh. 13 (1972), 767-772. English translation: Siberian Math. J. 13 (1973), 529-532. https://doi.org/10.1007/BF00971045
- V. Diskant, Strengthening of an isoperimetric inequality, Sibirskii Mat. Zh. 14 (1973), 873-877. English translation: Sib. Math. J. 14 (1973), 608-611.
- V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Math. 14 (1973), 1728-1731.
- H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), no. 6, 581-593. https://doi.org/10.2307/2313773
- M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225-1229. https://doi.org/10.1215/S0012-7094-83-05052-4
- X. Gao, A new reverse isoperimetric inequality and its stability, Math. Inequal. Appl. 15 (2012), no. 3, 733-743.
- R. Gardner, Geometric Tomography, Cambridge Univ. Press, New York, 1995.
- R. Gardner, The Brunn-Minkowski inequality, Minkowski's first inequality, and their duals, J. Math. Anal. Appl. 245 (2000), no. 2, 502-512. https://doi.org/10.1006/jmaa.2000.6774
- R. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), no. 3, 355-405. https://doi.org/10.1090/S0273-0979-02-00941-2
- M. Green and S. Osher, Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asian J. Math. 3 (1999), no. 3, 659-676. https://doi.org/10.4310/AJM.1999.v3.n3.a5
- H. Groemer, Geometric applications of Fourier series and spherical harmonics, Encyclo-pedia of Mathematics and its Applications, 61. Cambridge University Press, Cambridge, 1996.
- H. Groemer and R. Schneider, Stability estimates for some geometric inequalities, Bull. Lond. Math. Soc. 23 (1991), no. 1, 67-74. https://doi.org/10.1112/blms/23.1.67
- 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
- H. Hadwiger, Die isoperimetrische Ungleichung in Raum, Elem. Math. 3 (1948), 25-38.
- H. Hadwiger, Kurze Herleitung einer verscharften isoperimetrischen Ungleichung fur konvexe Korper, Rev. Fac. Sci. Univ. Istanbul, Ser. A 14 (1949), 1-6.
- H. Hadwiger, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer, Berlin, 1957.
- W. Y. Hsiang, An elementary proof of the isoperimetric problem, Chinese Ann. Math. 23 (2002), no. 1, 7-12.
- D. Klain, Bonnesen-type inequalities for surfaces of constant curvature, Adv. in Appl. Math. 39 (2007), no. 2, 143-154. https://doi.org/10.1016/j.aam.2006.11.004
- 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
- R. Osserman, Bonnesen-style isoperimetric inequality, Amer. Math. Monthly 86 (1979), no. 1, 1-29. https://doi.org/10.2307/2320297
- S. Pan, X. Tang, and X. Wang, A refined reverse isoperimetric inequality in the plane, Math. Inequal. Appl. 13 (2010), no. 2, 329-338.
- S. Pan and H. Xu, Stability of a reverse isoperimetric inequality, J. Math. Anal. Appl. 350 (2009), no. 1, 348-353. https://doi.org/10.1016/j.jmaa.2008.09.047
- D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
- J. R. Sangwine-Yager, Bonnesen-style inequalities for Minkowski relative geometry, Trans. Amer. Math. Soc. 307 (1988), no. 1, 373-382. https://doi.org/10.1090/S0002-9947-1988-0936821-5
- J. R. Sangwine-Yager, Mixe Volumes, Handbook of Covex Geometry, Vol. A, 43-71, Edited by P. Gruber & J. Wills, North-Holland, 1993.
- L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
- R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.
- Y. Xia, On reverse isoperimetric inequalities in two-dimensional space forms and related results, Math. Inequal. Appl. 18 (2015), no. 3, 1025-1032.
- W. Xu, J. Zhou, and B. Zhu, On containment measure and the mixed isoperimetric inequality, J. Inequal. Appl. 2013 (2103), 540, 11 pp. https://doi.org/10.1186/1029-242X-2013-11
- C. Zeng, L. Ma, J. Zhou, and F. Chen, The Bonnesen isoperimetric inequality in a surface of constant curvature, Sci. China Math. 55 (2012), no. 9, 1913-1919. https://doi.org/10.1007/s11425-012-4405-z
- C. Zeng, J. Zhou, and S. Yue, A symmetric mixed isoperimetric inequality for two planar convex domains, Acta Math. Sinica 55 (2012), no. 2, 355-362.
- G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95-116. https://doi.org/10.1112/S0025579300007208
- G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202. https://doi.org/10.4310/jdg/1214425451
- 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
- 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
- J. Zhou, Bonnesen-type inequalities, Acta Math. Sin. (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
- J. Zhou and F. Chen, The Bonnesen-type inequality in a plane of constant cuvature, J. Korean Math. Soc. 44 (2007), no. 6, 1363-1372. https://doi.org/10.4134/JKMS.2007.44.6.1363
- J. Zhou, Y. Du, and F. Cheng, Some Bonnesen-style inequalities for higher dimensions, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 12, 2561-2568. https://doi.org/10.1007/s10114-012-9657-6
- J. Zhou and D. Ren, Geometric inequalities from the viewpoint of integral geometry, Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 5, 1322-1339.
- J. Zhou, Y. Xia, and C. Zeng, Some new Bonnesen-style inequalities, J. Korean Math. Soc. 48 (2011), no. 2, 421-430. https://doi.org/10.4134/JKMS.2011.48.2.421
- J. Zhou, C. Zhou, and F. Ma, Isoperimetric deficit upper limit of a planar convex set, Rend. Circ. Mat. Palermo (2) 81 (2009), 363-367.