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Metric complexity is a Bryant--Tupper diversity

Gautam Aishwarya, Dongbin Li, Mokshay Madiman, Mark Meckes

Abstract

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.

Metric complexity is a Bryant--Tupper diversity

Abstract

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.

Paper Structure

This paper contains 4 sections, 8 theorems, 36 equations.

Table of Contents

  1. Introduction and main results
  2. The wedge sum and the proof of Theorem \ref{['thm: main']}
  3. Two properties of metric complexity
  4. Discussion

Key Result

Theorem 1.7

Let $(X, d)$ be a metric space, and $t > 0$. Then both and $C^t$ are diversities (in the sense of Definition def: BTdiv).

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 16 more