Abstract
Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, C., Leinster, T.: The Euler characteristic of a category as the sum of a divergent series. Homol. Homotopy Appl. 10, 41–51 (2008). http://intlpress.com/HHA/v10/n1/a3/
Biggins J.D., Bingham N.H.: Near-constancy phenomena in branching processes. Math. Proc. Camb. Phil. Soc. 110, 545–558 (1991)
Borceux F.: Handbook of Categorical Algebra 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications, vol. 50. Cambridge University Press, Cambridge (1994)
Borceux F.: Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51. Cambridge University Press, Cambridge (1994)
Falconer K.A.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2003)
Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Klain D.A., Rota G.-C.: Introduction to Geometric Probability. Cambridge University Press, Cambridge (1997)
Lawvere, F.W.: Metric spaces, generalized logic and closed categories. originally published in Rendiconti del Seminario Matematico e Fisico di Milano, vol. XLIII, pp. 135–166 (1973); republished in Reprints in Theory and Applications of Categories, vol. 1, pp. 1–37 (2002). http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html
Leinster, T.: The Euler characteristic of a category. Documenta Mathematica 13, 21–49 (2008). http://www.math.uni-bielefeld.de/documenta/vol-13/02.html
Leinster, T.: Metric spaces, post at The n-Category Café (2008, Feb) http://golem.ph.utexas.edu/category/2008/02/metric_spaces.html
Leinster, T.: A maximum entropy theorem with applications to the measurement of biodiversity. arXiv preprint. http://arxiv.org/abs/0910.0906v4
Leinster, T.: The magnitude of metric spaces. arXiv preprint. http://arxiv.org/abs/1012.5857v3
Mac Lane S.: Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics, vol. 5. Cambridge University Press, Cambridge (1998)
Meckes, M.: Positive definite metric spaces. Positivity, to appear. http://arxiv.org/abs/1012.5863v3
Miller, P.D.: Applied Asymptotic Analysis. Graduate Studies in Mathematics, vol. 75. American Mathematical Society (2006)
Solow A., Polasky S.: Measuring biological diversity. Environ. Ecol. Stat. 1, 95–107 (1994)
Speyer, D.: Re: Metric Spaces, reply to [11] (2008, Feb). http://golem.ph.utexas.edu/category/2008/02/metric_spaces.html#c014950
Willerton, S.: Heuristic and computer calculations for the magnitude of metric spaces. arXiv preprint. http://arxiv.org/abs/0910.5500v1
Willerton, S.: On the magnitude of spheres, surfaces and other homogeneous spaces. arXiv preprint. http://arxiv.org/abs/1005.4041v1
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leinster, T., Willerton, S. On the asymptotic magnitude of subsets of Euclidean space. Geom Dedicata 164, 287–310 (2013). https://doi.org/10.1007/s10711-012-9773-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-012-9773-6