A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.
Heuristic and computer calculations for the magnitude of metric spaces
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abstract
The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper more evidence is presented for a conjectured relationship with a geometric measure theoretic valuation. Firstly, a heuristic is given for deriving this valuation by considering 'large' subspaces of Euclidean space and, secondly, numerical approximations to the magnitude are calculated for squares, disks, cubes, annuli, tori and Sierpinski gaskets. The valuation is seen to be very close to the magnitude for the convex spaces considered and is seen to be 'asymptotically' close for some other spaces.
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UNVERDICTED 2representative citing papers
Introduces tractable metric spaces, characterizes them, and proves continuity of magnitude on this restricted class, including new proofs for subsets of R.
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Magnitude of metric measure spaces and integrals over geodesics
A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.
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Tractable Metric Spaces and Magnitude Continuity
Introduces tractable metric spaces, characterizes them, and proves continuity of magnitude on this restricted class, including new proofs for subsets of R.