《山东大学学报(理学版)》 ›› 2025, Vol. 60 ›› Issue (12): 94-102.doi: 10.6040/j.issn.1671-9352.0.2024.009
兰雨,冉启伟,曾春娜*
LAN Yu, RAN Qiwei, ZENG Chunna*
摘要： 通过运用凸函数的积分不等式获得了凸体的第i阶平均曲率积分任意次幂的估计,进而获得Chen不等式和Ros不等式的加强形式,并对Willmore不等式及Ros不等式进行了推广并获得了曲率熵不等式的上界估计。
中图分类号:
| [1] GAGE M E. An isoperimetric inequality with applications to curveshortening[J]. Duke Mathematical Journal, 1983, 50(4):1225-1229. [2] GREEN M, OSHER S. Steiner polynomials, wulff flows, and some new isoperimetric inequalities for convex plane curves[J]. Asian Journal of Mathematics, 1999, 3(3):659-676. [3] GAO L Y, PAN S L, TSAI D H. Nonlocal flow driven by the radius of curvature with fixed curvature integral[J]. The Journal of Geometric Analysis, 2020, 30(3):2939-2973. [4] LIN Y C, TSAI D H. Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves[J]. Journal of Evolution Equations, 2012, 12(4):833-854. [5] YANG Y L. Nonsymmetric extension of the green-osher inequality[J]. Geometriae Dedicata, 2019, 203(1):155-161. [6] YANG Y L, ZHANG D Y. The green-osher inequality in relativegeometry[J]. Bulletin of the Australian Mathematical Society, 2016, 94(1):155-164. [7] CARMO M P. Differential geometry of curves and surfaces:revised and updated[M]. 2nd ed. New York: Courier Dover Publications, 2016. [8] WILLMORE T J. Mean curvature of immersed surfaces[J]. Topics in Differential Geometry, 1976:149-156. [9] CHEN Bangyen. Geometry of submanifolds and its applications[J]. Science University of Tokyo, 1981:96. [10] CHEN B Y. On the total curvature of immersed manifolds, I: an inequality of fenchel-borsuk-willmore[J]. American Journal of Mathematics, 1971, 93(1):148-162. [11] ROS A. Compact hypersurfaces with constant scalar curvature and a congruence theorem[J]. Journal of Differential Geometry, 1988, 27(2):215-220. [12] 周家足,姜德烁,李明,等. 超曲面的Ros定理[J]. 数学学报,2009,52(6):1075-1084. ZHOU Jiazu, JIANG Deshuo, LI Ming, et al. On Ros theorem for hypersurfaces[J]. Acta Mathematica Sinica(Chinese Series), 2009, 52(6):1075-1084. [13] OSSERMAN R. Curvature in the eighties[J]. American Mathematical Monthly, 1990, 97(8):731-736. [14] ZHOU J Z. On the Willmore s theorem for convex hypersurfaces[J]. Acta Mathematica Scientia, 2011, 31(2):361-366. [15] SCHNEIDER R. Convex bodies: the Brunn-Minkowski theory[M]. Cambridge: Cambridge University Press, 2013. [16] STEINER J. Ueber parallele flächen[J]. Monatsbericht Preuss Akad Wiss, 1840, 2:114-118. [17] CIFRE M A H, ALONSO-GUTIÉRREZ D. Estimates for the integrals of powered i-th mean curvatures[M] //BIANCHI G, COLESANTI A, GRONCHI P, et al. Springer INdAM Series. Cham: Springer International Publishing, 2017:19-37. [18] GAO Laiyuan, PAN Shengliang, YANG Yunlong. Some notes on Green-Osher s inequality[J]. Journal of Mathematical Inequalities, 2015(2):369-380. [19] GUAN P F, NI L. Entropy and a convergence theorem for Gauss curvature flow in high dimension[J]. Journal of the European Mathematical Society, 2017, 19(12):3735-3761. [20] ZENG C N, DONG X, WANG Y L, et al. The log-Minkowski inequality of curvature entropy for non-symmetric convex bodies[EB/OL]. 2022: 2211.14484. https://arxiv.org/abs/2211.14484v1. |
| [1] | 李世龙,刘茜. 融合风险比率的双触发巨灾看跌期权及其定价[J]. 《山东大学学报(理学版)》, 2025, 60(3): 12-21. |
| [2] | 马丽娜 王国俊. Lukasiewicz三值逻辑中命题的真度值之集在［0,1］上的分布[J]. J4, 2009, 44(10): 54-59. |
|
