Metric complexity is a Bryant--Tupper diversity
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912976260300800 |
|---|---|
| author | Aishwarya, Gautam Li, Dongbin Madiman, Mokshay Meckes, Mark |
| author_facet | Aishwarya, Gautam Li, Dongbin Madiman, Mokshay Meckes, Mark |
| contents | The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superadditive for compact subsets of the real line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric complexity is a Bryant--Tupper diversity Aishwarya, Gautam Li, Dongbin Madiman, Mokshay Meckes, Mark Metric Geometry Information Theory 51F99, 94A17, 54E35 The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superadditive for compact subsets of the real line. |
| title | Metric complexity is a Bryant--Tupper diversity |
| topic | Metric Geometry Information Theory 51F99, 94A17, 54E35 |
| url | https://arxiv.org/abs/2507.09698 |