Geometriae Dedicata

, Volume 164, Issue 1, pp 287–310 | Cite as

On the asymptotic magnitude of subsets of Euclidean space

Original Paper

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.

Keywords

Magnitude Metric space Euler characteristic Intrinsic volumes 

Mathematics Subject Classification

28A75 18D20 52A23 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
  2. 2.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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