The Dual Orlicz–Aleksandrov–Fenchel Inequality
In this paper, the classical dual mixed volume of star bodies <inline-formula><math display="inline"><semantics><mrow><mover accent="true"><mi>V</mi><mo>˜</mo></mover><mrow><mo>(</mo><msub><mi&...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/11/2005 |
| _version_ | 1797548323812409344 |
|---|---|
| author | Chang-Jian Zhao |
| author_facet | Chang-Jian Zhao |
| author_sort | Chang-Jian Zhao |
| collection | DOAJ |
| description | In this paper, the classical dual mixed volume of star bodies <inline-formula><math display="inline"><semantics><mrow><mover accent="true"><mi>V</mi><mo>˜</mo></mover><mrow><mo>(</mo><msub><mi>K</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>K</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it <i>Orlicz multiple dual mixed volume.</i> We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of <inline-formula><math display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-dual multiple mixed volumes and <inline-formula><math display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established. |
| first_indexed | 2024-03-10T14:57:45Z |
| format | Article |
| id | doaj.art-4b5fa6f23e0647788f8c2f4b1a5e8bad |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T14:57:45Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-4b5fa6f23e0647788f8c2f4b1a5e8bad2023-11-20T20:26:27ZengMDPI AGMathematics2227-73902020-11-01811200510.3390/math8112005The Dual Orlicz–Aleksandrov–Fenchel InequalityChang-Jian Zhao0Department of Mathematics, China Jiliang University, Hangzhou 310018, ChinaIn this paper, the classical dual mixed volume of star bodies <inline-formula><math display="inline"><semantics><mrow><mover accent="true"><mi>V</mi><mo>˜</mo></mover><mrow><mo>(</mo><msub><mi>K</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>K</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it <i>Orlicz multiple dual mixed volume.</i> We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of <inline-formula><math display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-dual multiple mixed volumes and <inline-formula><math display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.https://www.mdpi.com/2227-7390/8/11/2005dual mixed volumedual Aleksandrov–Fenchel inequalityOrlicz harmonic radial additionOrlicz dual mixed volumeOrlicz dual Minkowski inequalitydual Orlicz–Brunn–Minkowski theory |
| spellingShingle | Chang-Jian Zhao The Dual Orlicz–Aleksandrov–Fenchel Inequality Mathematics dual mixed volume dual Aleksandrov–Fenchel inequality Orlicz harmonic radial addition Orlicz dual mixed volume Orlicz dual Minkowski inequality dual Orlicz–Brunn–Minkowski theory |
| title | The Dual Orlicz–Aleksandrov–Fenchel Inequality |
| title_full | The Dual Orlicz–Aleksandrov–Fenchel Inequality |
| title_fullStr | The Dual Orlicz–Aleksandrov–Fenchel Inequality |
| title_full_unstemmed | The Dual Orlicz–Aleksandrov–Fenchel Inequality |
| title_short | The Dual Orlicz–Aleksandrov–Fenchel Inequality |
| title_sort | dual orlicz aleksandrov fenchel inequality |
| topic | dual mixed volume dual Aleksandrov–Fenchel inequality Orlicz harmonic radial addition Orlicz dual mixed volume Orlicz dual Minkowski inequality dual Orlicz–Brunn–Minkowski theory |
| url | https://www.mdpi.com/2227-7390/8/11/2005 |
| work_keys_str_mv | AT changjianzhao thedualorliczaleksandrovfenchelinequality AT changjianzhao dualorliczaleksandrovfenchelinequality |