refs.bib

@book{klain_geometric_1997,
   title =     {Introduction to {Geometric Probability}},
   author =    {Daniel A. Klain and Gian-Carlo Rota},
   publisher = {Cambridge University Press},
   isbn =      {0521596548,9780521596541,052159362X,9780521593625},
   year =      {1997},
   series =    {},
   edition =   {},
   volume =    {},
   url =       {}
}

@book{horn_matrix_2013,
	edition = {2},
	title = {Matrix {Analysis}},
	publisher = {Cambridge University Press},
	author = {Horn, Roger A. and Johnson, Charles R.},
	year = {2013},
	file = {Horn and Johnson - 2013 - Matrix Analysis.pdf:/Users/ssyl55/Zotero/storage/TMIH528D/Horn and Johnson - 2013 - Matrix Analysis.pdf:application/pdf}
}

@book{schneider_convex_2014,
	edition = {Second Expanded},
	series = {Encyclopedia of {Mathematics} and {Its} {Applications} 151},
	title = {Convex {Bodies}: {The} {Brunn}-{Minkowski} {Theory}},
	isbn = {978-1-107-60101-7},
	language = {English},
	publisher = {Cambridge University Press},
	author = {Schneider, Rolf},
	year = {2014}
}

@article{lawvere_metric_1973,
	title = {Metric spaces, generalized logic, and closed categories},
	volume = {43},
	issn = {1424-9294},
	url = {https://doi.org/10.1007/BF02924844},
	doi = {10.1007/BF02924844},
	abstract = {The analogy between dist (a, b)+dist (b, c)≥dist (a, c) and hom (A, B) ⊗ hom (B, C)→hom (A, C) is rigorously developed to display many general results about metric spaces as consequences of a «generalized pure logic» whose «truth-values» are taken in an arbitrary closed category.},
	language = {en},
	number = {1},
	urldate = {2020-03-03},
	journal = {Rendiconti del Seminario Matematico e Fisico di Milano},
	author = {Lawvere, F. William},
	month = dec,
	year = {1973},
	pages = {135--166},
	file = {Springer Full Text PDF:/Users/ssyl55/Zotero/storage/P6HYGIUN/Lawvere - 1973 - Metric spaces, generalized logic, and closed categ.pdf:application/pdf}
}

@article{meckes_positive_2013,
	title = {Positive definite metric spaces},
	volume = {17},
	url = {arXiv:1012.5863},
	pages = {733--757},
	journal = {Positivity},
	year = {2013},
	author = {Meckes, Mark},
	file = {Meckes - Positive definite metric spaces.pdf:/Users/ssyl55/Zotero/storage/KPGFZ7BM/Meckes - Positive definite metric spaces.pdf:application/pdf}
}

@article{meckes_magnitude_2015,
	title = {Magnitude, {Diversity}, {Capacities}, and {Dimensions} of {Metric} {Spaces}},
	volume = {42},
	issn = {1572-929X},
	url = {https://doi.org/10.1007/s11118-014-9444-3},
	doi = {10.1007/s11118-014-9444-3},
	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.},
	language = {en},
	number = {2},
	urldate = {2020-02-27},
	journal = {Potential Analysis},
	author = {Meckes, Mark W.},
	month = feb,
	year = {2015},
	pages = {549--572},
	file = {Springer Full Text PDF:/Users/ssyl55/Zotero/storage/N7Q8DUZL/Meckes - 2015 - Magnitude, Diversity, Capacities, and Dimensions o.pdf:application/pdf}
}

@article{meckes_magnitude_2019,
	title = {On the magnitude and intrinsic volumes of a convex body in {E}uclidean space},
	url = {http://arxiv.org/abs/1904.08923},
	abstract = {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 closely related to intrinsic volumes. Here we prove an upper bound for the magnitude of a convex body in Euclidean space in terms of its intrinsic volumes. The result is deduced from an analogous known result for magnitude in \${\textbackslash}ell\_1{\textasciicircum}N\$, via approximate embeddings of Euclidean space into high-dimensional \${\textbackslash}ell\_1{\textasciicircum}N\$ spaces. As a consequence, we deduce a sufficient condition for infinite-dimensional subsets of a Hilbert space to have finite magnitude. The upper bound is also shown to be sharp to first order for an odd-dimensional Euclidean ball shrinking to a point; this complements recent work investigating the asymptotics of magnitude for large dilatations of sets in Euclidean space.},
	journaltitle = {{arXiv}:1904.08923 [math]},
	journal = {Mathematika},
	author = {Meckes, Mark W.},
	urldate = {2019-10-22},
	date = {2019-05-15},
	year = {2019},
	eprinttype = {arxiv},
	eprint = {1904.08923},
	keywords = {Mathematics - Functional Analysis, Mathematics - Metric Geometry},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/YXTUYU7A/Meckes - 2019 - On the magnitude and intrinsic volumes of a convex.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/2HD7NH67/1904.html:text/html}
}

@article{barcelo_magnitudes_2016,
	title = {On the magnitudes of compact sets in {E}uclidean spaces},
	url = {http://arxiv.org/abs/1507.02502},
	abstract = {The notion of the magnitude of a metric space was introduced by Leinster in [8] and developed in [10], [9], [11] and [16], but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton [9] in the special case of balls in dimension three. In general we show that the magnitude of an odd-dimensional ball is a rational function of its radius. In addition to Fourier-analytic and {PDE} techniques, the arguments also involve some combinatorial considerations.},
	journaltitle = {{arXiv}:1507.02502 [math]},
	journal = {Amer. J. Math},
	author = {Barcelo, Juan Antonio and Carbery, Anthony},
	urldate = {2019-11-08},
	date = {2016-07-13},
	year = {2016},
	eprinttype = {arxiv},
	eprint = {1507.02502},
	keywords = {51F99, 35J40, 31B15, 42B99, 05A10, Mathematics - Metric Geometry},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/AC4K73AA/Barcelo and Carbery - 2016 - On the magnitudes of compact sets in Euclidean spa.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/WVAUUXSE/1507.html:text/html}
}

@article{gimperlein_magnitude_2017,
	title = {On the magnitude function of domains in {Euclidean} space},
	url = {http://arxiv.org/abs/1706.06839},
	abstract = {},
	urldate = {2020-02-27},
	journal = {To appear in {A}mer. {J}. {M}ath.},
	author = {Gimperlein, Heiko and Goffeng, Magnus},
	month = nov,
	year = {2017},
	note = {arXiv: 1706.06839.},
	keywords = {Mathematics - Analysis of PDEs, Mathematics - Classical Analysis and ODEs, Mathematics - Metric Geometry, Mathematics - Spectral Theory},
	annote = {Comment: 16 pages, 3 figures},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/LQ3SXUGR/Gimperlein and Goffeng - 2017 - On the magnitude function of domains in Euclidean .pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/J2LGJP7P/1706.html:text/html}
}

@misc{weisstein_catalan_nodate,
	title = {{Wolfram Mathworld} {C}atalan {N}umber},
	url = {http://mathworld.wolfram.com/CatalanNumber.html},
	howpublished = {http://mathworld.wolfram.com/CatalanNumber.html},
	note = {Accessed: 2020-02-25},
	type = {Text},
	author = {Weisstein, Eric W.},
	urldate = {2020-02-25},
	langid = {english},
	file = {Snapshot:/Users/ssyl55/Zotero/storage/C4MR8A2S/CatalanNumber.html:text/html}
}

@misc{weisstein_double_nodate,
	title = {{Wolfram Mathworld} {D}ouble {F}actorial},
	url = {http://mathworld.wolfram.com/DoubleFactorial.html},
	howpublished = {http://mathworld.wolfram.com/DoubleFactorial.html},
	note = {Accessed: 2020-02-25},
	type = {Text},
	author = {Weisstein, Eric W.},
	urldate = {2020-02-25},
	langid = {english},
	file = {Snapshot:/Users/ssyl55/Zotero/storage/IMTVE47M/DoubleFactorial.html:text/html}
}

@misc{weisstein_legendre_nodate,
	title = {{Wolfram Mathworld} {L}egendre {D}uplication {F}ormula},
	url = {http://mathworld.wolfram.com/LegendreDuplicationFormula.html},
	howpublished = {http://mathworld.wolfram.com/LegendreDuplicationFormula.html},
	note = {Accessed: 2020-06-01},
	type = {Text},
	author = {Weisstein, Eric W.},
	urldate = {20202-06-01},
	langid = {english}
}

@article{willerton_heuristic_2009,
	title = {Heuristic and computer calculations for the magnitude of metric spaces},
	url = {http://arxiv.org/abs/0910.5500},
	abstract = {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 more evidence is presented for a conjectured relationship with a geometric measure theoretic valuation. Firstly, a heuristic is given for deriving this valuation by considering 'large' subspaces of Euclidean space and, secondly, numerical approximations to the magnitude are calculated for squares, disks, cubes, annuli, tori and Sierpinski gaskets. The valuation is seen to be very close to the magnitude for the convex spaces considered and is seen to be 'asymptotically' close for some other spaces.},
	urldate = {2020-02-27},
	journal = {arXiv},
	author = {Willerton, Simon},
	month = oct,
	year = {2009},
	note = {arXiv: 0910.5500 [math.MG]},
	keywords = {Mathematics - Category Theory, Mathematics - Metric Geometry},
	annote = {Comment: 16 pages, several figures and graphs. Version with slightly better quality graphics available from my homepage},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/JA6MRC9H/Willerton - 2009 - Heuristic and computer calculations for the magnit.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/QSPCI3UG/0910.html:text/html}
}

@article{willerton_magnitude_2014,
	title = {On the magnitude of spheres, surfaces and other homogeneous spaces},
	volume = {168},
	issn = {1572-9168},
	url = {https://doi.org/10.1007/s10711-013-9831-8},
	doi = {10.1007/s10711-013-9831-8},
	abstract = {In this paper we calculate the magnitude of metric spaces using measures rather than finite subsets as had been done previously. An explicit formula for the magnitude of an \$\$n\$\$-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold.},
	language = {en},
	number = {1},
	urldate = {2020-02-27},
	journal = {Geometriae Dedicata},
	author = {Willerton, Simon},
	month = feb,
	year = {2014},
	pages = {291--310},
	file = {Springer Full Text PDF:/Users/ssyl55/Zotero/storage/X6UCAYZN/Willerton - 2014 - On the magnitude of spheres, surfaces and other ho.pdf:application/pdf}
}

@article{willerton_magnitude_2017,
	title = {The magnitude of odd balls via {H}ankel determinants of reverse {B}essel polynomials},
	url = {http://arxiv.org/abs/1708.03227},
	abstract = {Magnitude is an invariant of metric spaces with origins in enriched category theory. Using potential theoretic methods, Barcel{\textbackslash}'o and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schr{\textbackslash}"oder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.},
	journaltitle = {{arXiv}:1708.03227 [math]},
	journal = {arXiv},
	author = {Willerton, Simon},
	urldate = {2019-10-22},
	date = {2017-08-10},
	year = {2017},
	note = {arXiv:1708.03227 [math.MG]},
	eprinttype = {arxiv},
	eprint = {1708.03227},
	keywords = {Mathematics - Metric Geometry, Mathematics - Classical Analysis and {ODEs}},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/JZ95AKE5/Willerton - 2017 - The magnitude of odd balls via Hankel determinants.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/H3J56MP9/1708.html:text/html}
}

@article{leinster_euler_2006,
	title = {The {Euler} characteristic of a category},
	url = {http://arxiv.org/abs/math/0610260},
	abstract = {The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula for the cardinality of the colimit of a diagram of sets is proved, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Mobius-Rota inversion from posets to categories.},
	urldate = {2020-02-27},
	journal = {Documenta Mathematica 13 (2008)},
	author = {Leinster, Tom},
	month = oct,
	year = {2006},
	note = {arXiv.CT/0610260},
	keywords = {Mathematics - Algebraic Topology, Mathematics - Category Theory, Mathematics - Combinatorics},
	annote = {Comment: 24 pages},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/4P7GG5SP/Leinster - 2006 - The Euler characteristic of a category.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/N6JMB4LQ/0610260.html:text/html}
}

@article{leinster_magnitude_2011,
	title = {The magnitude of metric spaces},
	url = {http://arxiv.org/abs/1012.5857},
	abstract = {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 general categorical definition that clarifies the analogies between various cardinality-like invariants in mathematics. Although this motivation is a world away from geometric measure, magnitude, when applied to subsets of R{\textasciicircum}n, turns out to be intimately related to invariants such as volume, surface area, perimeter and dimension. We describe several aspects of this relationship, providing evidence for a conjecture (first stated in {arXiv}:0908.1582) that magnitude subsumes all the most important invariants of classical integral geometry.},
	journaltitle = {{arXiv}:1012.5857 [math]},
	journal = {Documenta Mathematica},
	author = {Leinster, Tom},
	urldate = {2019-10-22},
	date = {2011-01-22},
	year = {2011},
	eprinttype = {arxiv},
	eprint = {1012.5857},
	keywords = {Mathematics - Metric Geometry, Mathematics - Category Theory, Mathematics - General Topology, Mathematics - Geometric Topology},
	file = {arXiv Fulltext PDF:/Users/ssyl55/Zotero/storage/BRHHFK46/Leinster - 2011 - The magnitude of metric spaces.pdf:application/pdf;arXiv.org Snapshot:/Users/ssyl55/Zotero/storage/PS5FGBE3/1012.html:text/html}
}

@article{leinster_asymptotic_2013,
	title = {On the asymptotic magnitude of subsets of {Euclidean} space},
	volume = {164},
	issn = {1572-9168},
	url = {https://doi.org/10.1007/s10711-012-9773-6},
	doi = {10.1007/s10711-012-9773-6},
	abstract = {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.},
	language = {en},
	number = {1},
	urldate = {2020-02-27},
	journal = {Geometriae Dedicata},
	author = {Leinster, Tom and Willerton, Simon},
	month = jun,
	year = {2013},
	pages = {287--310},
	file = {Springer Full Text PDF:/Users/ssyl55/Zotero/storage/F2KRZCWB/Leinster and Willerton - 2013 - On the asymptotic magnitude of subsets of Euclidea.pdf:application/pdf}
}

@article{leinster_magnitude_2017,
	address = {Berlin, Boston},
	title = {The magnitude of a metric space: from category theory to geometric measure theory},
	isbn = {978-3-11-055083-2},
	shorttitle = {The magnitude of a metric space},
	url = {https://www.degruyter.com/view/books/9783110550832/9783110550832-005/9783110550832-005.xml},
	urldate = {2020-02-27},
	journal = {Measure {Theory} in {Non}-{Smooth} {Spaces}},
	publisher = {De Gruyter},
	author = {Leinster, Tom and Meckes, Mark W.},
	year = {2017},
	doi = {10.1515/9783110550832-005},
	file = {Full Text PDF:/Users/ssyl55/Zotero/storage/2KRJLX4B/Leinster and Meckes - 2017 - The magnitude of a metric space from category the.pdf:application/pdf}
}