| Inequalities relating to Lp-version of Petty＇s conjectured projection inequality |
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| 引用本文： | 王卫东,冷岗松. Inequalities relating to Lp-version of Petty＇s conjectured projection inequality[J]. 应用数学和力学(英文版), 2007, 28(2): 269-276. DOI: 10.1007/s 10483-007-0216-x |
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| 作者姓名： | 王卫东 冷岗松 |
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| 作者单位： | [1]Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China; [2]Department of Mathematics, Hubei Institute for Nationalities, Enshi 445000, Hubei Province, P. R. China |
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| 基金项目： | 国家自然科学基金;湖北省教育厅重点自然科学研究基金 |
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| 摘 要： | Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.
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| 关 键 词： | 不等式 多维空间几何 单一性 几何方程 |
| 收稿时间： | 2005-06-27 |
| 修稿时间： | 2006-11-30 |
Inequalities relating to Lp-version of Petty’s conjectured projection inequality |
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| Wang Wei-dong,Leng Gang-song. Inequalities relating to Lp-version of Petty’s conjectured projection inequality[J]. Applied Mathematics and Mechanics(English Edition), 2007, 28(2): 269-276. DOI: 10.1007/s 10483-007-0216-x |
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| Authors: | Wang Wei-dong Leng Gang-song |
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| Affiliation: | (1) Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China;(2) Department of Mathematics, Hubei Institute for Nationalities, Enshi, 445000, Hubei Province, P. R. China |
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| Abstract: | Petty’s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to L p -version of the Petty’s conjectured projection inequality is developed by using the notions of the L p -mixed volume and the L p -dual mixed volume, the relation of the L p -projection body and the geometric body Γ−p K, the Bourgain-Milman inequality and the L p -Busemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Γ−p K, the reverses of L p -version of the Petty’s conjectured projection inequality and the L p -Petty projection inequality are given, respectively. Project supported by the National Natural Science Foundation of China (No.10671117), the Key Science Research Foundation of Education Department of Hubei Province of China (No.2003A005). |
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| Keywords: | Lp-version Petty projection inequality Petty's conjectured projection inequality Lp-projection body reverse |
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