On the magnitude of spheres, surfaces and other homogeneous spaces

Willerton, Simon

doi:10.1007/s10711-013-9831-8

Cited by 24 publications

(64 citation statements)

References 11 publications

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“…Another proof of Corollary 4.3 can be given using [19,Proposition 3.2.3]. As an application of Corollary 4.3, we obtain the magnitude of the length ℓ ternary Cantor set C ℓ (see [23,Theorem 10], [41,Theorem 4]):…”

Section: Magnitude In Normed Spacesmentioning

confidence: 98%

“…and there is a similar formula for odd n; see [41,Theorem 7]. Lemma 3.10 is particularly useful in analyzing the magnitude function of a homogeneous space A, since it implies that tA possesses a weight measure for every t > 0, which is moreover independent of t (up to normalization).…”

Section: 1mentioning

confidence: 99%

“…A first nontrivial example is a compact interval [a, b] ⊆ R. A straightforward computation (see [41,Theorem 2]) shows that…”

Section: 1mentioning

confidence: 99%

“…In [41], a definition was proposed for the magnitude of certain compact metric spaces, and connections were found between magnitude and some intrinsic volumes of Riemannian manifolds. Shortly thereafter, the paper [19] appeared which laid out for the first time the general theory of the magnitude of finite metric spaces; and [27] which put the asymptotic approach of [23] for studying magnitude of compact spaces on firm footing, and showed that it also coincides with the definition used in [41].…”

Section: Introductionmentioning

confidence: 99%

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

Leinster¹,

Meckes²

2017

Measure Theory in Non-Smooth Spaces

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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 to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper gives an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities. Some new results are proved, including a geometric formula for the magnitude of a convex body in ℓ n 1 .

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Section: Magnitude In Normed Spacesmentioning

confidence: 98%

Section: 1mentioning

confidence: 99%

“…A first nontrivial example is a compact interval [a, b] ⊆ R. A straightforward computation (see [41,Theorem 2]) shows that…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Leinster¹,

Meckes²

2017

Measure Theory in Non-Smooth Spaces

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“…In the metric context, the meaning of magnitude becomes clearer after one extends the definition from finite to compact spaces (which is done by approximating them by finite subspaces). Magnitude for compact metric spaces has recognizable geometric content: for example, the magnitude of a 3-dimensional ball is a cubic polynomial in its radius [2,Theorem 2], and the magnitude of a homogeneous Riemannian manifold is closely related to its total scalar curvature [41,Theorem 11].…”

Section: Open Questionsmentioning

confidence: 99%

Maximizing Diversity in Biology and Beyond

Leinster

Meckes

2016

Entropy

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Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold (2012) proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry.

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On the asymptotic magnitude of subsets of Euclidean space

Leinster

Willerton

2012

Geom Dedicata

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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 Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

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On the magnitude of spheres, surfaces and other homogeneous spaces

Cited by 24 publications

References 11 publications

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

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

Maximizing Diversity in Biology and Beyond

On the asymptotic magnitude of subsets of Euclidean space

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