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Metric spaces
"... These slides: available on my web pageExecutive summary Magnitude is a realvalued invariant of metric spaces. It seems not to have been previously investigated. Conjecturally, it captures a great deal of geometric information. It arose from a general study of ‘size ’ in mathematics. Plan 1. Where d ..."
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Cited by 10 (3 self)
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These slides: available on my web pageExecutive summary Magnitude is a realvalued invariant of metric spaces. It seems not to have been previously investigated. Conjecturally, it captures a great deal of geometric information. It arose from a general study of ‘size ’ in mathematics. Plan 1. Where does magnitude come from? 2. The magnitude of a finite space 3. The magnitude of a compact space
On the magnitude of spheres, surfaces and other homogeneous spaces
, 2010
"... In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier results. An explicit formula for the magnitude of an nsphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannia ..."
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Cited by 4 (1 self)
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In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier results. An explicit formula for the magnitude of an nsphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold. In the particular case of a homogeneous surface the form of the asymptotics can be given exactly up to vanishing terms and this involves just the area and Euler characteristic in the way
Spread: a measure of the size of metric spaces
, 2014
"... Motivated by LeinsterCobbold 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 lin ..."
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Motivated by LeinsterCobbold 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 scaledependent dimension is introduced and seen, numerically, to be close to the Hausdorff dimension for approximations