Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces

Abstract

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural and direct manner than in previous works by Leinster, Willerton, and the author. The new definition uncovers a previously unknown relationship between magnitude and capacities of sets. Exploiting this relationship, it is shown that for a compact subset of Euclidean space, the magnitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension.2010 Mathematics Subject Classification. Primary 51F99; Secondary 31B15, 49Q15. Key words and phrases. Magnitude of metric spaces; capacity; Minkowski dimension. 1 Finite metric spacesWe now recall the definition of the magnitude of a finite metric space. Definition 2.1. Given a finite metric space (A, d), define the matrix ζ ∈ R A×A by ζ(a, b) := e −d(a,b) . A vector w ∈ R A is a weighting for A if for each a ∈ A, (2.1) (ζw)(a) = b∈A e −d(a,b) w(b) = 1.

The magnitude of a metric space: from category theory to geometric measure theory

“…The lower bound was proved in [19,Theorem 3.5.6] for p = 1 and [27,Theorem 4.5] for the general case. 1 The proof here follows the approach used in [28] for ℓ n 2 (see Proposition 5.6 and the remarks following Corollary 5.3 there).…”

“…Let k = R, and define |·| : [0, ∞] → R to be either the indicator function of [0, 1] or that of [0, 1). It is shown in Section 8 of [28] that these are essentially the only possibilities for |·|, and that the resulting magnitude of a finite ultrametric space is simply the number of balls of radius 1 (closed or open, respectively) needed to cover it. It is also shown that this leads naturally to the notion of ε-entropy or ε-capacity.…”

“…Nevertheless, for both aesthetic and practical reasons, it is desirable to be able to work directly with A itself, as opposed to approximations of A by finite subspaces. Two different more direct approaches to defining magnitude for compact positive definite spaces were developed in [27,28]. In essence, these papers introduced two different topologies on the space w ∈ R A supp w is finite .…”

“…It will therefore come as no surprise that Fourier analysis is our key tool in that setting. This approach was developed in [28] for ℓ n 2 , but with some additional effort we can work not only with more general norms but with the broader class of p-(quasi)norms for 0 < p ≤ 1.…”

“…The paper [28] introduced yet another equivalent approach to magnitude for compact spaces, which makes magnitude more accessible to a wide variety of analytic techniques. Using a result from potential theory, [28] showed in particular that magnitude can be used to recover the Minkowski dimension of a compact set in Euclidean space.…”