A study on preservation of parallel pairs and triangle equality attainment pairs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913936375283712 |
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| author | Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya |
| author_facet | Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya |
| contents | This paper investigates the structure and preservation of parallel pairs and triangle equality attainment (TEA) pairs in normed linear spaces. We begin by providing functional characterizations of these pairs in normed linear spaces and provide a characterization of finite-dimensional Banach spaces with numerical index one. We then study parallel pairs and TEA pairs in the $p$-direct sum of normed linear spaces. Next, we show that in finite-dimensional Banach spaces, a bijective bounded linear operator preserves TEA pairs if and only if it preserves parallel pairs, and this equivalence remains valid in finite-dimensional polyhedral spaces for operators with rank greater than one. We further explore some geometric consequences related to this preservation and provide a characterization of isometries on certain polyhedral Banach spaces as norm-one bijective operators that preserve the norm at extreme points and also preserve the parallel or TEA pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study on preservation of parallel pairs and triangle equality attainment pairs Manna, Jayanta Mandal, Kalidas Paul, Kallol Sain, Debmalya Functional Analysis Primary 46B20, Secondary 47L05 This paper investigates the structure and preservation of parallel pairs and triangle equality attainment (TEA) pairs in normed linear spaces. We begin by providing functional characterizations of these pairs in normed linear spaces and provide a characterization of finite-dimensional Banach spaces with numerical index one. We then study parallel pairs and TEA pairs in the $p$-direct sum of normed linear spaces. Next, we show that in finite-dimensional Banach spaces, a bijective bounded linear operator preserves TEA pairs if and only if it preserves parallel pairs, and this equivalence remains valid in finite-dimensional polyhedral spaces for operators with rank greater than one. We further explore some geometric consequences related to this preservation and provide a characterization of isometries on certain polyhedral Banach spaces as norm-one bijective operators that preserve the norm at extreme points and also preserve the parallel or TEA pairs. |
| title | A study on preservation of parallel pairs and triangle equality attainment pairs |
| topic | Functional Analysis Primary 46B20, Secondary 47L05 |
| url | https://arxiv.org/abs/2507.07774 |