참고문헌
- T. F. Banchoff and W. F. Pohl, A generalization of the isoperimetric inequality, J. Differential Geometry 6 (1971/72), 175-192. https://doi.org/10.4310/jdg/1214430403
- J. Bokowski and E. Heil, Integral representations of quermassintegrals and Bonnesen- style inequalities, Arch. Math. (Basel) 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 Korper, Springer-Verlag, Berlin- New York, 1974.
- O. Bottema, Eine obere Grenze fur das isoperimetrische Dezit ebener Kurven, Nederl. Akad. Wetensch. Proc. A66 (1933), 442-446.
- Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer-Verlag, Berlin, Heidelberg, 1988.
- C. Croke, A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), no. 2, 187-192. https://doi.org/10.1007/BF02566344
- V. Diskant, A generalization of Bonnesen's inequalities, Dokl. Akad. Nauk SSSR 213 (1973), 519-521.
-
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. - E. Grinberg, D. Ren, and J. Zhou, The symetric isoperimetric deficit and the contain- ment problem in a plan of constant curvature, preprint.
- E. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies, Math. Ann. 291 (1991), no. 1, 75-86. https://doi.org/10.1007/BF01445191
- E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77-115. https://doi.org/10.1112/S0024611599001653
- 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
- G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambradge Univ. Press, Cambradge/New York, 1951.
- R. Howard, The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2779-2787. https://doi.org/10.1090/S0002-9939-98-04336-6
- C. C. Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220. https://doi.org/10.2307/1970287
- W. Y. Hsiung, An elementary proof of the isoperimetric problem, Chinese Ann. Math. Ser. A 23 (2002), no. 1, 7-12.
- H. Ku, M. Ku, and X. Zhang, Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math. 49 (1997), no. 6, 1162-1187. https://doi.org/10.4153/CJM-1997-057-x
- 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. https://doi.org/10.1007/s11425-010-4018-3
- E. Lutwak, D. Yang, and G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17-38. https://doi.org/10.4310/jdg/1090425527
- 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 inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1-29. https://doi.org/10.2307/2320297
- A. Pleijel, On konvexa kurvor, Nordisk Math. Tidskr. 3 (1955), 57-64.
- G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
- D. Ren, Topics in Integral Geometry, World Scientific, Sigapore, 1994.
- L. A. Santalo, Integral Geometry and Geometric Probability, Reading, MA: Addison- Wesley, 1976.
- A. Stone, On the isoperimetric inequality on a minimal surface, Calc. Var. Partial Differential Equations 17 (2003), no. 4, 369-391. https://doi.org/10.1007/s00526-002-0174-9
- D. Tang, Discrete Wirtinger and isoperimetric type inequalities, Bull. Austral. Math. Soc. 43 (1991), no. 3, 467-474. https://doi.org/10.1017/S0004972700029312
- E. Teufel, A generalization of the isoperimetric inequality in the hyperbolic plane, Arch. Math. (Basel) 57 (1991), no. 5, 508-513. https://doi.org/10.1007/BF01246751
- E. Teufel, Isoperimetric inequalities for closed curves in spaces of constant curvature, Results Math. 22 (1992), no. 1-2, 622-630. https://doi.org/10.1007/BF03323109
- S. Wei and M. Zhu, Sharp isoperimetric inequalities and sphere theorems, Pacific J. Math. 220 (2005), no. 1, 183-195. https://doi.org/10.2140/pjm.2005.220.183
- J. L. Weiner, A generalization of the isoperimetric inequality on the 2-sphere, Indiana Univ. Math. J. 24 (1974/75), 243-248. https://doi.org/10.1512/iumj.1974.24.24021
- J. L. Weiner, Isoperimetric inequalities for immersed closed spherical curves, Proc. Amer. Math. Soc. 120 (1994), no. 2, 501-506. https://doi.org/10.1090/S0002-9939-1994-1163337-4
- 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. https://doi.org/10.24033/asens.1299
- G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183-202. https://doi.org/10.4310/jdg/1214425451
- G. Zhang and J. Zhou, Containment measures in integral geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006.
- 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, 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.
- 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.
-
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. https://doi.org/10.2307/2154603 -
J. Zhou, When can one domain enclose another in
$R^{3}$ ?, J. Austral. Math. Soc. Ser. A 59 (1995), no. 2, 266-272. https://doi.org/10.1017/S1446788700038660 - 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. https://doi.org/10.1090/S0002-9939-98-04369-X
- J. Zhou, On Willmore's inequality for submanifolds, Canad. Math. Bull. 50 (2007), no. 3, 474-480. https://doi.org/10.4153/CMB-2007-047-4
- 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.
-
J. Zhou, The Willmore functional and the containment problem in
$R^{4}$ , Sci. China Ser. A 50 (2007), no. 3, 325-333. https://doi.org/10.1007/s11425-007-0029-0 - J. Zhou, Bonnesen-type inequalities on the plane, Acta Math. Sinica (Chin. Ser.) 50 (2007), no. 6, 1397-1402.
- 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. https://doi.org/10.4134/JKMS.2007.44.6.1363
피인용 문헌
- The Bonnesen isoperimetric inequality in a surface of constant curvature vol.55, pp.9, 2012, https://doi.org/10.1007/s11425-012-4405-z
- Reverse Bonnesen style inequalities in a surface $$\mathbb{X}_\varepsilon ^2$$ of constant curvature vol.56, pp.6, 2013, https://doi.org/10.1007/s11425-013-4578-0
- Some Bonnesen-style inequalities for higher dimensions vol.28, pp.12, 2012, https://doi.org/10.1007/s10114-012-9657-6
- ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT vol.50, pp.1, 2013, https://doi.org/10.4134/BKMS.2013.50.1.175
- On containment measure and the mixed isoperimetric inequality vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-540
- Bonnesen-style symmetric mixed inequalities vol.2016, pp.1, 2016, https://doi.org/10.1186/s13660-016-1146-5
- Bonnesen-style inequalities on surfaces of constant curvature vol.2018, pp.1, 2018, https://doi.org/10.1186/s13660-018-1918-1
- Reverse Bonnesen-style inequalities on surfaces of constant curvature vol.29, pp.06, 2018, https://doi.org/10.1142/S0129167X18500404