On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908606352326656 |
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| author | Bredies, Kristian Rodriguez, Jonathan Chirinos Naldi, Emanuele |
| author_facet | Bredies, Kristian Rodriguez, Jonathan Chirinos Naldi, Emanuele |
| contents | In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_14039 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces Bredies, Kristian Rodriguez, Jonathan Chirinos Naldi, Emanuele Functional Analysis 46A55 (Primary) 46N10, 52A05 (Secondary) In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory. |
| title | On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces |
| topic | Functional Analysis 46A55 (Primary) 46N10, 52A05 (Secondary) |
| url | https://arxiv.org/abs/2304.14039 |