[1209.2300v1] Spread: a measure of the size of metric spaces

Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate

Mathematics > Metric Geometry

arXiv:1209.2300v1 (math)
[Submitted on 11 Sep 2012 (this version), latest version 6 Jan 2015 (v2)]

Title:Spread: a measure of the size of metric spaces

Authors:Simon Willerton
View PDF
Abstract:Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster's magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen, numerically, to be close to the Hausdorff dimension for approximations to certain fractals.
Comments: 13 pages
Subjects: Metric Geometry (math.MG)
Cite as: arXiv:1209.2300 [math.MG]
  (or arXiv:1209.2300v1 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.1209.2300

Submission history

From: Simon Willerton [view email]
[v1] Tue, 11 Sep 2012 12:21:23 UTC (81 KB)
[v2] Tue, 6 Jan 2015 20:41:58 UTC (760 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
view license

Current browse context:

math.MG
< prev   |   next >
new | recent | 2012-09
Change to browse by:
math

References & Citations

1 blog link

(what is this?)

Bookmark

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)
archive