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The microscopic weighting on a metric space
Emily Roff, Simon Willerton
We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space. The microscopic weighting is obtained as th…
Is magnitude 'generically continuous' for finite metric spaces?
Hirokazu Katsumasa, Emily Roff, Masahiko Yoshinaga
Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the m…
The small-scale limit of magnitude and the one-point property
Emily Roff, Masahiko Yoshinaga
The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude c…
The maximum entropy of a metric space
Tom Leinster, Emily Roff
We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with…