2009/08/31 by Tom Leinster, Simon Willerton · 33 citations
Computer Science · Mathematics · #Cantor set #Cardinality (data modeling) #Digital Image Processing Techniques #Euclidean distance #Euclidean distance matrix #Euclidean geometry #Euclidean space #Finite set #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Magnitude (astronomy) #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Metric space #Pure mathematics #Real line #Seven-dimensional space #Space (punctuation) #math.CA #math.CT #math.MG #msc:18D20 #msc:28A75 #msc:54E35
paper · pdf · doi:10.1007/s10711-012-9773-6
published in Geometriae Dedicata 164(1), 287-310 (Springer Science+Business Media) · 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references added
arxiv created 2012/08/06 · openalex publication_date 2012/08/27 · arxiv updated 2013/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.