Positive definite metric spaces
@article{Meckes2010PositiveDM, title={Positive definite metric spaces}, author={Mark W. Meckes}, journal={Positivity}, year={2010}, volume={17}, pages={733-757} }
Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the…
48 Citations
On the magnitudes of compact sets in Euclidean spaces
- Mathematics
- 2015
abstract:The notion of the magnitude of a metric space was introduced by Leinster and developed in works by Leinster, Meckes and Willerton, but the magnitudes of familiar sets in Euclidean space are…
The magnitude of metric spaces
- MathematicsDocumenta Mathematica
- 2013
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a…
ON THE MAGNITUDE AND INTRINSIC VOLUMES OF A CONVEX BODY IN EUCLIDEAN SPACE
- MathematicsMathematika
- 2020
Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is…
Approximating the Convex Hull via Metric Space Magnitude
- Mathematics, Computer Science
- 2019
This paper restricts the sets to finite subsets of Euclidean space and investigates its individual components, and gives an explicit formula for the corrected inclusion-exclusion principle, and defines a quantity associated with each point, called the moment, which gives an intrinsic ordering to the points.
On the magnitude of odd balls via potential functions
- Mathematics
- 2018
Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in…
THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY
- Mathematics
- 2017
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out…
On the magnitude of a finite dimensional algebra
- Mathematics
- 2015
There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical…
The magnitude and spectral geometry
- Mathematics
- 2022
We study the geometric significance of Leinster’s notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in R2. For a large…
The maximum entropy of a metric space
- Mathematics, Computer ScienceArXiv
- 2019
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 a notion of…
On the asymptotic magnitude of subsets of Euclidean space
- MathematicsGeometriae Dedicata
- 2012
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…
References
SHOWING 1-10 OF 48 REFERENCES
The magnitude of metric spaces
- Mathematics
- 2010
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a…
Metric spaces and positive definite functions
- Mathematics
- 1938
As poo we get the space Em with the distance function maxi-, ... I xi X. Let, furthermore, lP stand for the space of real sequences with the series of pth powers of the absolute values convergent.…
Heuristic and computer calculations for the magnitude of metric spaces
- Mathematics
- 2009
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…
On the asymptotic magnitude of subsets of Euclidean space
- Mathematics
- 2013
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…
Embeddings and Extensions in Analysis
- Mathematics
- 1975
I. Isometric Embedding.- 1. Introduction.- 2. Isometric Embedding in Hilbert Space.- 3. Functions of Negative Type.- 4. Radial Positive Definite Functions.- 5. A Characterization of Subspaces of Lp,…
Hyperbolic spaces are of strictly negative type
- Mathematics
- 2001
We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the…
Does negative type characterize the round sphere
- Mathematics
- 2007
We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact…
Geometric Nonlinear Functional Analysis
- Mathematics
- 1999
Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets…