Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants
In this paper, we firstly investigate the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-01-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/14/1/72 |
| _version_ | 1797490118536200192 |
|---|---|
| author | Bingren Chen Qi Liu Yongjin Li |
| author_facet | Bingren Chen Qi Liu Yongjin Li |
| author_sort | Bingren Chen |
| collection | DOAJ |
| description | In this paper, we firstly investigate the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> closely related to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>, along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> explicitly. |
| first_indexed | 2024-03-10T00:26:24Z |
| format | Article |
| id | doaj.art-26815d6d096d4d9fa4eb356728e0a568 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T00:26:24Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-26815d6d096d4d9fa4eb356728e0a5682023-11-23T15:32:58ZengMDPI AGSymmetry2073-89942022-01-011417210.3390/sym14010072Inscribed Triangles in the Unit Sphere and a New Class of Geometric ConstantsBingren Chen0Qi Liu1Yongjin Li2School of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaSchool of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaSchool of Mathematics, Sun Yat-sen University, Guangzhou 510275, ChinaIn this paper, we firstly investigate the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> closely related to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>, along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> explicitly.https://www.mdpi.com/2073-8994/14/1/72Banach spacesgeometric constantsuniformly non-square |
| spellingShingle | Bingren Chen Qi Liu Yongjin Li Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants Symmetry Banach spaces geometric constants uniformly non-square |
| title | Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants |
| title_full | Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants |
| title_fullStr | Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants |
| title_full_unstemmed | Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants |
| title_short | Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants |
| title_sort | inscribed triangles in the unit sphere and a new class of geometric constants |
| topic | Banach spaces geometric constants uniformly non-square |
| url | https://www.mdpi.com/2073-8994/14/1/72 |
| work_keys_str_mv | AT bingrenchen inscribedtrianglesintheunitsphereandanewclassofgeometricconstants AT qiliu inscribedtrianglesintheunitsphereandanewclassofgeometricconstants AT yongjinli inscribedtrianglesintheunitsphereandanewclassofgeometricconstants |