Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (1): 215-227.doi: 10.1007/s10473-025-0117-7
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Haizhong Li1, Yao Wan2
Received:2024-09-04
Published:2025-02-06
About author:Haizhong Li, E-mail,: lihz@tsinghua.edu.cn; Yao Wan, E-mail,: yaowan@cuhk.edu.hk
Supported by:CLC Number:
Haizhong Li, Yao Wan. STABILITY OF THE ISOPERIMETRIC INEQUALITY IN HYPERBOLIC PLANE[J].Acta mathematica scientia,Series B, 2025, 45(1): 215-227.
| [1] Andrews B, Classification of limiting shapes for isotropic curve flows. J Amer Math Soc, 2003, 16(2): 443-459 [2] Andrews B, Chen X Z, Wei Y. Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space. J Eur Math Soc, 2021, 23(7): 2467-2509 [3] Andrews B, Hu Y X, Li H Z. Harmonic mean curvature flow and geometric inequalities. Adv Math, 2020, 375: 107393 [4] Andrews B, Wei Y. Quermassintegral preserving curvature flow in hyperbolic space. Geom Funct Anal, 2018, 28(5): 1183-1208 [5] Ball K. An elementary introduction to modern convex geometry. Flavors of Geometry. Math Sci Res Inst Publ, 1997, 31: 1-58 [6] Blaschke W. Über affine geometrie I: Isoperimetrische eigenschaften von ellipse und ellipsoid. Ber Verh Sächs Akad Leipzig Math Phys Kl, 1916, 68: 217-239 [7] Ball K, Böröczky K. Stability of some versions of the Prékopa-Leindler inequality. Monatsh Math, 2011, 163(1): 1-14 [8] Bögelein V, Duzaar F, Scheven C. A sharp quantitative isoperimetric inequality in hyperbolic n-space. Calc Var Partial Differential Equations, 2015, 54(4): 3967-4017 [9] Borisenko A, Miquel V. Total curvatures of convex hypersurfaces in hyperbolic space. Illinois Journal of Mathematics-ILL J MATH, 1999, 43(1): 61-78 [10] Borisenko A, Miquel V. Comparison Theorems on Convex Hypersurfaces in Hadamard Manifolds. Annals of Global Analysis and Geometry, 2002, 21: 191-202 [11] Böröczky K. Stability of the Blaschke-Santaló and the affine isoperimetric inequality. Adv Math, 2010, 225(4): 1914-1928 [12] Böröczky K, Makai Jr E. On the volume product of planar polar convex bodies-upper estimates: the polygonal case and stability. In preparation [13] Brendle S, Guan P F, Li J F. An inverse curvature type hypersurface flow in space forms. 2019, private note [14] Burago Y, Zalgaller V A. Geometric Inequalities. Berlin: Springer-Verlag, 1988 [15] Cabezas-Rivas E, Miquel V. Volume preserving mean curvature flow in the hyperbolic space. Indiana Univ Math J, 2006, 56(5): 2061-2086 [16] Gao C Q, Zhou R. Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space. Nonlinear Anal, 2024, 244: Art 113537 [17] Ge Y X, Wang G F, Wu J. Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II. J Differential Geom, 2014, 98(2): 237-260 [18] Chang M. Bonnesen-style inequalities on surfaces of constant curvature. J Inequal Appl, 2018: Art 325 [19] Chou K S, Wang X J. The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205(1): 33-83 [20] Hu Y X, Li H Z. Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space. Calc Var Partial Differential Equations, 2019, 58(2): Art 55 [21] Hu Y X, Li H Z. Blaschke-Santaló type inequalities and quermassintegral inequalities in space forms. Adv Math, 2023, 413: Art 108826 [22] Hu Y X, Li H Z, Wei Y. Locally constrained curvature flows and geometric inequalities in hyperbolic space. Math Ann, 2022, 382(3/4): 1425-1474 [23] Hu Y X, Wei Y, Zhou T L. A Heintze-Karcher type inequality in hyperbolic space. J Geom Anal, 2024, 34(4): Art 113 [24] Huang Y, Lutwak E, Yang D, Zhang G Y. Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math, 2016, 216(2): 325-388 [25] Huang Y, Zhao Y M. On the Lp dual Minkowski problem. Adv Math, 2018, 332: 57-84 [26] Hug D, Lutwak E, Yang D, Zhang G Y. On the Lp Minkowski problem for polytopes. Discrete Comput Geom, 2005, 33(4): 699-715 [27] Ivaki M. On the stability of the p-affine isoperimetric inequality. J Geom Anal, 2014, 24(4): 1898-1911 [28] Ivaki M. Stability of the Blaschke-Santaló inequality in the plane. Monatsh Math, 2015, 177(3): 451-459 [29] John F. Polar correspondence with respect to a convex region. Duke Math J, 1937, 3(2): 355-369 [30] Klain D. Bonnesen-type inequalities for surfaces of constant curvature. Adv in Appl Math, 2007, 39(2): 143-154 [31] Li H Z, Wan Y. Classification of solutions for the planar isotropic Lp dual Minkowski problem. arXiv:2209.14630 [32] Li H Z, Wan Y. The Christoffel problem in the hyperbolic plane. Adv in Appl Math, 2023, 150: Art 102557 [33] Li H Z, Wan Y, Xu B T. The discrete horospherical p-Minkowski problem in hyperbolic space. Adv Math, 2024, 453: Art 109851 [34] Li H Z, Wei Y, Xiong C W. A geometric inequality on hypersurface in hyperbolic space. Adv Math, 2014, 253: 152-162 [35] Li H Z, Xu B T. Hyperbolic p-sum and horospherical p-Brunn-Minkowski theory in hyperbolic space. arXiv:2211.06875 [36] Lutwak E. Dual mixed volumes. Pacific J Math, 1975, 58(2): 531-538 [37] Lutwak E. The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38(1): 131-150 [38] Lutwak E. The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas. Adv Math, 1996, 118(2): 244-294 [39] Lutwak E, Yang D, Zhang G Y. On the Lp-Minkowski problem. Trans Amer Math Soc, 2004, 356(11): 4359-4370 [40] Lutwak E, Yang D, Zhang G Y. Lp dual curvature measures. Adv Math, 2018, 329: 85-132 [41] Makowski M. Mixed volume preserving curvature flows in hyperbolic space. arXiv:1208.1898 [42] Palmer B, Perdomo O. Rotating drops with helicoidal symmetry. Pacific J Math, 2015, 273(2): 413-441 [43] Perdomo O. Embedded constant mean curvature hypersurfaces on spheres. Asian J Math, 2010, 14(1): 73-108 [44] Petty C M. Affine isoperimetric problems. Discrete Geometry and Convexity, 1985, 440: 113-127 [45] Saint-Raymond J. Sur le volume des corps convexes symétriques. Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ Math Univ Pierre et Marie Curie, vol. 46, Univ Paris VI, Paris, 1981, pp. Exp. No. 11, 25 (French) [46] Sahjwani P, Scheuer J. Stability of the quermassintegral inequalities in hyperbolic space. J Geom Anal, 2024, 34(1): Art 13 [47] Santaló L A. An affine invariant for convex bodies of n-dimensional space. Portugal Math, 1949, 8: 155-161 [48] Schmidt E. Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl. Math Z, 1943, 49: 1-109 [49] Schmidt E. Der Brunn-Minkowskische Satze und sein Spiegeltheorem sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und hyperbolischen Geometrie. Math Ann, 1948, 120: 307-422 [50] Schmidt E. Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I. Math Nachr, 1948, 1: 81-157 [51] Rolf Schneider. Convex Bodies: the Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 2014 [52] Wang G F, Xia C. Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space. Adv Math, 2014, 259: 532-556 [53] Werner E, Ye D P. New Lp affine isoperimetric inequalities. Adv Math, 2008, 218(3): 762-780 [54] Zeng C N, Ma L, Zhou J Z, Chen F W. The Bonnesen isoperimetric inequality in a surface of constant curvature. Sci China Math, 2012, 55(9): 1913-1919 [55] Zhou J Z, Chen F W. The Bonnesen-type inequalities in a plane of constant curvature. J Korean Math Soc, 2007, 44(6): 1363-1372 [56] Zhou R, Zhou T L. Stability of alexandrov-fenchel type inequalities for nearly spherical sets in space forms. arXiv:2306.02581 |
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