Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes
Our main aim is to generalize the classical mixed volume V(K1,…,Kn) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call it...
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Wiley
2018-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2018/9752178 |
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| author | Chang-Jian Zhao |
| author_facet | Chang-Jian Zhao |
| author_sort | Chang-Jian Zhao |
| collection | DOAJ |
| description | Our main aim is to generalize the classical mixed volume V(K1,…,Kn) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call it Orlicz multiple mixed volume of convex bodies K1,…,Kn, and Ln, denoted by Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of Lp-multiple mixed volume Vp(K1,…,Kn,Ln) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals. |
| format | Article |
| id | doaj-art-b4d6100ea8cf4a8e8e2934b337ab58f5 |
| institution | OA Journals |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
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| series | Journal of Function Spaces |
| spelling | doaj-art-b4d6100ea8cf4a8e8e2934b337ab58f52025-08-20T02:23:31ZengWileyJournal of Function Spaces2314-88962314-88882018-01-01201810.1155/2018/97521789752178Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed VolumesChang-Jian Zhao0Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang, ChinaOur main aim is to generalize the classical mixed volume V(K1,…,Kn) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call it Orlicz multiple mixed volume of convex bodies K1,…,Kn, and Ln, denoted by Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of Lp-multiple mixed volume Vp(K1,…,Kn,Ln) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.http://dx.doi.org/10.1155/2018/9752178 |
| spellingShingle | Chang-Jian Zhao Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes Journal of Function Spaces |
| title | Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes |
| title_full | Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes |
| title_fullStr | Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes |
| title_full_unstemmed | Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes |
| title_short | Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes |
| title_sort | orlicz aleksandrov fenchel inequality for orlicz multiple mixed volumes |
| url | http://dx.doi.org/10.1155/2018/9752178 |
| work_keys_str_mv | AT changjianzhao orliczaleksandrovfenchelinequalityfororliczmultiplemixedvolumes |