Positive definite metric spaces

By Mark W. Meckes. 2010

82 citations recorded by the source corpus.

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 classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of $$\ell _p^n$$ for $$p \le 2$$ are proved, generalizing results of Leinster for $$p=1,2$$ using properties of these spaces which are somewhat stronger than positive definiteness.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2de3e6b6f0778cffdcd132e9315d93ceab1761ac

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Mark W. Meckes
2. Tom Leinster
3. Russell Lyons
4. Simon Willerton
5. Richard Hepworth
6. Michael Shulman
7. Aaron D. Lauda
8. E. Chang

In Mathematical Physics

1. Mark W. Meckes
2. Tom Leinster
3. Russell Lyons
4. Simon Willerton
5. Richard Hepworth
6. Michael Shulman
7. Alexei Borodin
8. Subhro Ghosh

Citations in the corpus

• The magnitude of a graph (2014, background): “Metric spaces provide a third context for magnitude [8, 10, 11].”
• The magnitude of metric spaces (2010, methodology): “The magnitude of spheres is especially well understood [22, 42, 26].”
• Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces (2013, methodology): “4 in [13]), and motivates the definition of |A| = ∞ when A does not possess a weighting.”
• Maximizing diversity in biology and beyond (2015, unknown): “By definition, the magnitude of a finite metric space X is the magnitude of the matrix Z = (e)i,j∈X ; see [13,38,45], for instance.”
• Distance covariance in metric spaces (2011, background): “, Naor (2010) and Deza and Laurent (1997).”