A complete characterization of Radon planes whose unit spheres are regular polygons
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2018
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916364372934656 |
|---|---|
| author | Mandal, Kalidas Sain, Debmalya Paul, Kallol |
| author_facet | Mandal, Kalidas Sain, Debmalya Paul, Kallol |
| contents | We study the structure of the unit sphere of polygonal Radon planes from a geometric point of view. In particular, we prove that a $ 2 $-dimensional real polygonal Banach space $ \mathbb{X} $ cannot be a Radon plane if the number of vertices of its unit sphere is $ 4n, $ for some $ n \in \mathbb{N}. $ We next obtain a complete characterization of polygonal Radon planes in terms of a tractable geometric concept introduced in this article. It follows from our characterization that every regular polygon with $ 4n+2 $ vertices, where $ n \in \mathbb{N}, $ is the unit sphere of a Radon plane. We further give example of a family of Radon planes for which the unit spheres are hexagons, but not regular ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_04760 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A complete characterization of Radon planes whose unit spheres are regular polygons Mandal, Kalidas Sain, Debmalya Paul, Kallol Functional Analysis Primary 46B20, Secondary 52A21 We study the structure of the unit sphere of polygonal Radon planes from a geometric point of view. In particular, we prove that a $ 2 $-dimensional real polygonal Banach space $ \mathbb{X} $ cannot be a Radon plane if the number of vertices of its unit sphere is $ 4n, $ for some $ n \in \mathbb{N}. $ We next obtain a complete characterization of polygonal Radon planes in terms of a tractable geometric concept introduced in this article. It follows from our characterization that every regular polygon with $ 4n+2 $ vertices, where $ n \in \mathbb{N}, $ is the unit sphere of a Radon plane. We further give example of a family of Radon planes for which the unit spheres are hexagons, but not regular ones. |
| title | A complete characterization of Radon planes whose unit spheres are regular polygons |
| topic | Functional Analysis Primary 46B20, Secondary 52A21 |
| url | https://arxiv.org/abs/1809.04760 |