| Some basic inequalities in higher dimensional non-Euclid space |
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| 引用本文： | Ding-hua YANG College of Mathematics and Software Sciences,Sichuan Normal University,Chengdu 610066,China, Chengdu Institute of Computer Applications,Chinese Academy of Sciences,Chengdu 610041,China. Some basic inequalities in higher dimensional non-Euclid space[J]. 中国科学A辑(英文版), 2007, 50(3): 423-438. DOI: 10.1007/s11425-007-0011-x |
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| 作者姓名： | Ding-hua YANG College of Mathematics and Software Sciences Sichuan Normal University Chengdu 610066 China Chengdu Institute of Computer Applications Chinese Academy of Sciences Chengdu 610041 China |
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| 作者单位： | Ding-hua YANG College of Mathematics and Software Sciences，Sichuan Normal University，Chengdu 610066，China; Chengdu Institute of Computer Applications，Chinese Academy of Sciences，Chengdu 610041，China |
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| 基金项目： | 国家重点基础研究发展计划(973计划) |
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| 摘 要： | In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.
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| 收稿时间： | 2005-05-30 |
| 修稿时间： | 2006-09-11 |
Some basic inequalities in higher dimensional non-Euclid space |
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| Ding-hua Yang. Some basic inequalities in higher dimensional non-Euclid space[J]. Science in China(Mathematics), 2007, 50(3): 423-438. DOI: 10.1007/s11425-007-0011-x |
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| Authors: | Ding-hua Yang |
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| Affiliation: | College of Mathematics and Software Sciences, Sichuan Normal University, Chengdu 610066, China;Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610041, China |
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| Abstract: | In this paper, the concept of a finite mass-points system ΣN(H(A))(N > n) being in a sphere in an n-dimensional hyperbolic space H n and a finite mass-points system ΣN(S(A))(N > n) being in a hyperplane in an n-dimensional spherical space S n is introduced, then, the rank of the Cayley-Menger matrix-ΛN(H) (or a-ΛN(S)) of the finite mass-points system ΣN(S(A)) (or ΣN(S(A))) in an n-dimensional hyperbolic space H n (or spherical space S n) is no more than n + 2 when ΣN(H(A))(N > n) (or ΣN(S(A))(N > n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang’s inequalities, the Neuberg-Pedoe’s inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space H n and in an n-dimensional spherical space S n are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought. This work was partially supported by the National Key Basic Research Project of China (Grant No. 2004CB318003). |
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| Keywords: | higher dimensional Euclid space simplex volume radius of circumscribed sphere metric addition Yang-Zhang's inequality Neuberg-Pedoe's inequality |
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