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April 21, 2010

On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Posted by Simon Willerton

Back in my recent post on intrinsic volumes I said that I was just finishing a paper on the magnitude of certain Riemannian manifolds. It is now essentially finished.

I will put it on the arxiv next week: comments and suggestions are welcome.

There are two main results in the paper. The first is a calculation of the magnitude of the nn -sphere and the second is the asymptotic behaviour of the magnitude for an arbitrary homogeneous Riemannian manifold in terms of volume and total scalar curvature, a special case of the latter is the asymptotic behaviour of the magnitude of a homogeneous surface. I will explain these below, but first remind you a little about magnitude.

On the Magnitude…

Magnitude of finite metric spaces (under the name cardinality) was introduced by Tom Leinster at this very café, by analogy with his notion of Euler characteristic of a category. In two posts, Entropy, Diversity and Cardinality (Part 1) and Part 2, he explained how this was related to ideas in ecology and went on to prove a result on how magnitude helps you maximize biological diversity. In another direction Tom and I defined the magnitude of various infinite subsets of Euclidean space by approximating these with finite metric spaces. In the case of a finite metric space, as the space is scaled up, the magnitude gets closer and closer to the number of points: in the case of some nice subsets of Euclidean space we observed that as the subset is scaled up, the magnitude seems to tend to some linear combination of classical ‘intrinsic volumes’ of the subset.

[If you want some idea of the intuition I use to think of the magnitude then you should look at the post More Magnitude of Metric Spaces and Problems with Penguins.]

In this paper I take a different tack in defining the magnitude of infinite metric spaces, one that was suggested to me by Tom Leinster and by David Speyer. This involves using a (signed) measure on a metric space, and it clearly generalizes the idea of having a weight associated to each point in a finite metric space. Using this approach gives the same answer as we got in examples we’d previously calculated.

A key observation, essentially made by David Speyer here, is that for a homogeneous space XX (i.e., one that looks the same at every point) the magnitude can be calculated using any measure μ\mu by the formula

|X|= xXdμ xXe d(x,y)dμ,for any yX\left|X\right|= \frac{\int_{x\in X} d\mu}{\int_{x\in X}e^{-d(x,y)} d\mu},\qquad\text{for any} y\in X

provided that the denominator is finite and non-zero. Aside from working with subsets of the line, this is currently the only way I know of calculating the magnitude of infinite metric spaces.

…of Spheres,…

There are two obvious metrics to consider on the nn-sphere: there’s the ‘intrinsic’ metric, coming from thinking of it as an abstract Riemannian manifold and there’s the ‘Euclidean subspace’ metric, coming from thinking of it as a subset of n+1\mathbb{R}^{n+1}. On the Earth, the intrinsic distance between Sheffield and Riverside is how far I would have to travel over the surface of the Earth to get from one to the other; whereas the Euclidean subspace distance is how far I would have to travel through the Earth if I dug a straight-line tunnel from one to the other. In previous work, Tom and I concentrated on spaces with the subspace metric, but here I am concerned with the intrinsic metric.

It is a very fun exercise to use Speyer’s formula to explicitly calculate the magnitude for S R nS_R^n, the nn-sphere of radius RR with its intrinsic metric (or at least I found it fun). This is done in the paper and gives the following answer.

|S R n|={2((Rn1) 2+1)((Rn3) 2+1)((R1) 2+1)1+e πR for n even πR((Rn1) 2+1)((Rn3) 2+1)((R2) 2+1)1e πR for n odd. \left|S^n_R\right|=\begin{cases} \frac{2 \Big(\Big(\frac{R}{n-1}\Big)^2 + 1\Big)\Big(\Big(\frac{R}{n-3}\Big)^2 + 1\Big) \dots \Bigl(\Bigl(\frac{R}{1}\Bigr)^2 + 1\Bigr) }{1+e^{-\pi R}}& \text{for n even}\\ \frac{\pi R \Bigl(\Bigl(\frac{R}{n-1}\Bigr)^2 + 1\Bigr)\Bigl(\Bigl(\frac{R}{n-3}\Bigr)^2 + 1\Bigr) \dots \Bigl(\Bigl(\frac{R}{2}\Bigr)^2 + 1\Bigr) }{1-e^{-\pi R}}\quad& \text{for n odd}. \end{cases}

The denominator rapidly tends to one as RR increases. So the large, or asymptotic, behaviour of the magnitude is governed by the numerator which, for fixed dimension nn, is a polynomial in RR with some nice properties. For instance the leading order and constant terms can be identified in terms of classical invariants, namely the volume and the Euler characteristic, so, writing ω n\omega_n for the volume of the unit nn-ball, the numerator is the following form:

1n!ω nvol(S 1 n)R n++χ(S 1 n).\tfrac{1}{n!\omega_n}vol(S^n_1)R^n +\dots+\chi(S^n_1).

This is the kind of thing that Tom and I had been expecting for spaces with the Euclidean subspace metric. However, we will see below that the other term we can identify differs from what we naively expect in the Euclidean metric case.

One obvious question to ask here is the following.

  • What is the magnitude of the nn-sphere with the subspace metric?

…Surfaces…

All closed, compact surfaces (spheres, tori, etc.) can be equipped with a homogeneous Riemannian metric. For the sphere this should be clear as it is naturally thought of as a homogeneous space. Also the torus is commonly thought of as the quotient of the plane by an integer lattice, making it homogeneous. Similarly every higher genus surface can be obtained by quotienting the hyperbolic plane with an appropriate group of isometries, analogous to 2\mathbb{Z}^2.

For a surface with a homogeneous metric we can use Speyer’s formula from above to calculate the magnitude. I haven’t been able to find any nice closed form for the magnitude of higher genus surfaces as in the case of spheres, but there’s a nice computation giving the asymptotic behaviour. For a metric space XX and t>0t\gt0 we can write tXt X for XX with the metric scaled up by a factor of tt. The for a surface Σ\Sigma with a homogeneous metric, as tt\to \infty

|tΣ| =12πArea(tΣ)+χ(tΣ)+O(t 2) =12πArea(Σ)t 2+χ(Σ)+O(t 2). \begin{aligned}\left|t\Sigma\right|&= \tfrac{1}{2\pi}Area(t\Sigma)+\chi(t\Sigma) +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}). \end{aligned}

This is precisely the kind of behaviour Tom and I had conjectured for subspaces of Euclidean spaces.

There are four main ingredients in my proof of this result: one is Speyer’s formula for the magnitude of a homogeneous metric space mentioned above, another is Watson’s Lemma, a staple of asymptotic analyis, and the other two are properties of Gauss curvature mentioned in my post on intrinsic volumes, namely that the Gauss curvature determines the circumference of small circles (at least to low order), and that the integral of the Gauss curvature is proportional to the Euler characteristic.

If we are calculating |tΣ|\left|t\Sigma\right| using Speyer’s formula, then the key thing is to understand the denominator; so for any fixed x 0Σx_0\in \Sigma we want to consider

xΣe td(x,x 0)dx.\int_{x\in\Sigma} e^{-t d(x,x_0)}d x.

If tt is very large then because of the negative exponential, parts of Σ\Sigma which are not very close to x 0x_0 will only give exponentially small contributions. This rigorous way to state this is Watson’s lemma, but the point is that for large tt the dominant contributions will come from local information, i.e., curvature type information.

Writing F rF_r for the subset of Σ\Sigma at a distance of rr from x 0x_0 we can rewrite the the above integral as follows

xΣe td(x,x 0)dx= r0e trLength(F r)dr.\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=\int_{r\ge 0}e^{-t r} Length(F_r) d r.

But we know that for small rr the length, or circumference, of F rF_r is determined at low order by the Gauss curvature K(x 0)K(x_0).

Length(F r)=2πr(116K(x 0)r 2+O(r 4))as r0.Length(F_r)=2\pi r \big(1 - \tfrac{1}{6}K(x_0)r^{2}+O(r^{4})\big)\qquad{as} r\to 0.

Using this, together with Watson’s Lemma we get that the integral has the form

xΣe td(x,x 0)dx=2πt 2(1K(x 0)t 2+O(t 4))as t.\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big) \qquad{as} t\to\infty.

This means that as tt\to \infty we find, as we wanted, that

|tΣ| = xΣdx xΣe td(x,x 0)dx= xΣdx2πt 2(1K(x 0)t 2+O(t 4)) =12πt 2 xΣdx+12π xΣK(x 0)dx+O(t 2) =12πArea(Σ)t 2+χ(Σ)+O(t 2), \begin{aligned} \left|t\Sigma\right| &=\frac{\int_{x\in\Sigma} d x}{\int_{x\in\Sigma} e^{-t d(x,x_0)}d x} =\frac{\int_{x\in\Sigma} d x}{2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big)}\\ &=\tfrac{1}{2\pi}t^2\int_{x\in\Sigma} d x + \tfrac{1}{2\pi}\int_{x\in\Sigma} K(x_0)d x +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}), \end{aligned}

by the fact that the Gauss curvature is constant on a homogeneous space and that the integral of the Gauss curvature is proportional to the Euler characteristic.

…and Other Homogeneous Spaces

For an arbitrary homogeneous Riemannian manifold XX of dimension nn, you can use the same technique as for surfaces to calculate the leading order terms of the asymptotics of the magnitude. The key point is to identify the relevant analogues of the Gauss curvature and the Euler characteristic. These turn out to be the scalar curvature and the total scalar curvature.

At every point xx of a Riemannian manifold XX there is a scalar curvature τ(x)\tau(x) which specializes to twice the Gauss curvature in the case of a surface. This has the property of the Gauss curvature that at a point xx the volume of the set of points F rF_r a distance rr from xx is, for small rr, determined to leading terms by the scalar curvature:

vol n1(F r)=r n1σ n1(1τ(x)r 26n+)as r0.vol_{n-1}(F_r)= r^{n-1}\sigma_{n-1}\left(1-\tfrac{\tau(x)r^2}{6n}+\dots\right)\quad \text{as} r\to 0.

The total scalar curvature, tsc(X)tsc(X), is defined to be the integral of the scalar curvature;

tsc(X) Xτ(x)dx,tsc(X)\coloneqq\int_X \tau(x) d x,

The total scalar curvature seems to crop up in various parts of mathematics and physics, but I don’t know much about that. The connection with intrinsic volumes is that the total scalar curvature is, for closed manifolds, the next non-trivial intrinsic volume after the volume:

μ n2(X)=14πtsc(X).\mu_{n-2}(X) = \frac{1}{4\pi}tsc(X).

In the case that XX is a surface (so n=2n=2) we recover the relationship between the total Gauss curvature and the Euler characteristic

Running through through the same analysis as for the case of surfaces above, you find that for XX a homogeneous Riemannian manifold the magnitude looks asymptotically as follows:

|tX|=1n!ω nvol(X)t n+n+16n!ω ntsc(X)t n2+O(t n4).\left|t X\right| = \tfrac{1}{n!\omega_n}vol(X)t^n+\tfrac{n+1}{6n!\omega_n}tsc(X)t^{n-2}+O(t^{n-4}).

There are a few questions that jump out at this point.

  • Can these methods be used to calculate further terms in the asymptotic expansion for homogeneous manifolds?
  • Are these leading terms the same for the asymptotic behaviour of non-homogeneous Riemannian manifolds?

I should probably note here that the asymptotic behaviour is only one part of the picture and it would also be good to know what is going on for ‘small’ spaces: what is the magnitude measuring?

Posted at April 21, 2010 7:33 AM UTC

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31 Comments & 1 Trackback

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Thanks for the post and paper. I’m a big fan of this magnitude of metric spaces stuff.

Once every couple of years I pick up Street’s paper “Algebra of oriented simplexes” hoping my math kung fu has progressed enough for me to finally understand it. Coincidentally, the last time I picked it up recently made me think of this magnitude of metric spaces stuff. Probably for no good reason other than lack of sleep though. Let me throw some thoughts out there and see if anything makes sense…

I liked the idea of relating the 1-cocycle condition to a composite triangle in a category, i.e.

[012]=[12][02]+[01]=0\partial[012] = [12]-[02]+[01] = 0

rearranges to

[02]=[01]+[12][02] = [01]+[12]

which is interpreted as a commuting triangle.

Similarly

[0123]=[123][023]+[013][012]=0\partial[0123] = [123]-[023]+[013]-[012] = 0

rearranges to

[123]+[013]=[023]+[012].[123]+[013] = [023]+[012].

The pictures that go with this are great.

Then I had a whacky idea. What if we turn the square brackets into magnitudes instead, i.e.

|02|=|01|+|12|.|02| = |01|+|12|.

This says the magnitude of [02] equals the sum of the magnitudes of [01] and [12]. But if magnitude is “volume” then we are talking about lengths of the sides of a triangle. When is the sum of two lengths of a triangle equal to the length of the third side? When the triangle is degenerate, i.e. has no area. Area is zero? Boundary is zero? Could they be related?

This idea bumps up trivially in dimension.

The 2-cocycle condition gives

|123|+|013|=|023|+|012||123|+|013| = |023|+|012|

which says the sum of areas of two triangles of a tetrahedron are equal to the sum of the areas of the two remaining triangles of the tetrahedron. When does that happen? It happens when the 3-simplex is degenerate, i.e. it has no volume.

The picture is kind of pretty. You partition the boundary of an nn-simplex into two (n1)(n-1)-complexes. The (n1)(n-1)-cocycle condition says the magnitude of these two complexes are equal.

We have magnitudes of metric spaces and magnitudes of categories, is there a such thing as a magnitude of a diagram in a category?

Apologies if this makes absolutely no sense and is completely irrelevant to this magnitude business :)

Posted by: Eric Forgy on April 21, 2010 4:14 PM | Permalink | Reply to this

First reactions

Because a diagram is a functor, it has a domain category, with its own magnitude; but the image is a subcategory, too (unless that notion is evil?), which again has its own magnitude. Maybe you want some combination of these two?

Posted by: some guy on the street on April 22, 2010 12:05 AM | Permalink | Reply to this

Re: First reactions

Hmm…

If you start with the category 1\mathbf{1} with two objects 0,1 and one non-identity morphism \to and a functor F:1CF:\mathbf{1}\to C with F(0)=F(1)F(0) = F(1), then the image of FF is not a subcategory of CC unless F()=1 F(0)F(\to) = 1_{F(0)}. Is that what you meant by evil?

I spent about a week on the n-Forum (starting around here) making these guys pull their hair out because I was making a mistake thinking the image of a functor was a category.

After all that, please do not now tell me the image of a functor is a category after all :)

What I was thinking about when I asked

We have magnitudes of metric spaces and magnitudes of categories, is there a such thing as a magnitude of a diagram in a category?

I had the magnitude of the image of a functor in mind (which need not be a category). More generally, I’d be interested if there was a magnitude for a general directed graph.

Now the admittedly weak link to magnitudes is that I was wondering if \partial might be considered a functor mapping categories (or diagrams or graphs) to magnitudes.

Posted by: Eric on April 22, 2010 1:54 AM | Permalink | Reply to this

Re: First reactions

Oops. I guess you guys call that category 2\mathbf{2} :)

Posted by: Eric on April 22, 2010 4:54 AM | Permalink | Reply to this

Re: First reactions

oh! hmm… it’s weirder than I took care to think about, this image-of-a-functor business. Of course, the identity morphism will be in there, but unless the image of \to is an involution… I really should try to remember these issues more carefully!

More carefully, we should probably take the right factorization of the diagram F:𝒟𝒞F:\mathcal{D}\rightarrow\mathcal{C} as 2Hom(0,1):=𝒞(F(0),F(1))𝒞| {F(0),F(1)}𝒞; \mathbf{2}\longrightarrow \Big\langle Hom(0,1)\mathllap{:}{\,=}\mathcal{C}(F(0),F(1))\Big\rangle \longrightarrow \mathcal{C}|_{\{F(0),F(1)\}} \longrightarrow \mathcal{C} ; being the factor functors forgetting only structure, only stuff, and only property, respectively — and wherein none of the intermediate categories are really the image of the functor where you wanted to define a magnitude. So… take or leave what you will.

Posted by: some guy on the street on April 22, 2010 7:21 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

One thing that immediately strikes me when looking at this is the similarity of all the definitions here to things that come up when using the energy method for bounding Hausdorff dimensions.

For instance the formula

(1) Xexp(d(x,y))dμ(y)=1 \int_X \exp(-d(x,y)) d\mu(y) = 1

is precisely the expression (in the language of the energy method, which of course is in the language of physics) that the potential felt by the point xx with mass distribution μ\mu and forces that decay exponentially is one.

Similarly the expression

(2)|X|= Xdμ(x)= X Xexp(d(x,y))dμ(x)dμ(y) |X| = \int_X d\mu(x) = \int_X\int_X \exp(-d(x,y)) d\mu(x)d\mu(y)

is precisely the statement that the magnitude of the space is the total mass required, or equivalently the total potential energy of the system.

This leads to a nice physical interpretation: that the magnitude of a metric space is exactly the amount of mass needed to be spread out over the metric space so that every point feel a total potential energy of one (when the potentials/forces decay exponentially).

I don’t think there’ll be any direct connection to Hausdorff measures though because everything there relies upon a potential which is singular at small distances, whereas this is “regularized” near zero distance and instead cares about large distances. Still the terminology might make what you are saying familiar to more people, which is always good!

A resource, probably not the best one, for this sort of stuff is the excellent book on Brownian Motion by Morters and Peres, a draft of which can be posted here: www.stat.berkeley.edu/~peres/bmbook.pdf. The pertinent information is in chapter 4 about page 109 (a section about Hausdorff measures and dimensions).

Posted by: Brent Werness on April 21, 2010 5:10 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Brent said:

This leads to a nice physical interpretation: that the magnitude of a metric space is exactly the amount of mass needed to be spread out over the metric space so that every point feel a total potential energy of one (when the potentials/forces decay exponentially).

I prefer to think in terms of something like ‘heat generation’ rather than mass because we actually allow signed measures (which I didn’t emphasize in the post before) and I don’t have a good intuition for negative mass. I described this point of view in the post More Magnitude of Metric Spaces and Problems with Penguins and I’ve now linked to this in the main text above.

The similarity with the Energy Method is intriguing. I wasn’t aware of it. The main difference, however, is that the Energy Method uses a power of the distance where we use the exponential. I would really like to find some connection with this fractal dimension ideas though, as the growth of the magnitude does seem connected with ‘dimension’ in some sense. So for instance in On the asymptotic magnitude of subsets of Euclidean space Tom and I show that the magnitude of the Cantor set asymptotically grows like the Hausdorff dimension log 32\log_3 2; asymptotically, the magnitude of the length tt Cantor set looks like

f(t)t log 32f(t)t^{\log_3 2}

for some ‘almost constant’, multiplicatively periodic function ff.

Posted by: Simon Willerton on April 22, 2010 1:58 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Typo? The denominator rapidly tends to zero as R increases … zero -> one

Posted by: Charlie C on April 21, 2010 6:07 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Fixed, thanks.

Posted by: Simon Willerton on April 21, 2010 7:13 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

In the Introduction you credit the knowledge that magnitude is always defined for finite subsets of Euclidean space partly to me. Actually I believe that was a combination of observations of Tom, David Corfield, Yemon Choi, and David Speyer. My contributions started with generalizing that away from Euclidean space.

Posted by: Mark Meckes on April 21, 2010 9:04 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I was basing that on my interpretation of Tom’s comments in the post An Adventure in Analysis. I will figure out how to rewrite this before putting it on the arxiv. Thanks.

I see you’re giving some seminars on magnitude. I hope they’re going well!

Posted by: Simon Willerton on April 21, 2010 11:45 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

They seem to be going well. You’ve released this draft just in time for me to talk about it in my last talk tomorrow!

Posted by: Mark Meckes on April 22, 2010 1:24 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Josh Shadlen should also be added to that list.

Posted by: Tom Leinster on April 21, 2010 11:48 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

It’s interesting to me that you say the measure-based approach to defining magnitude was suggested to you by Tom and David S., whereas your own thoughts had been more along the lines of currents. To me the measure approach is the obvious first thing to look at, but prompted by some things Tom has said to me I now think something like currents might be the way to go.

In any case, I think it’s clear that whenever a weight measure exists, the magnitude should be exactly as you define it. I hope that finding the right class of measure-like things to consider will provide some more versatile tools for proving that weight measures exist for many spaces.

Posted by: Mark Meckes on April 22, 2010 3:08 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark said:

prompted by some things Tom has said to me I now think something like currents might be the way to go.

Do you want to expand on this? My reasons were mainly that on the one hand I had never done any measure theory – I certainly never took a course on it – and on the other hand I was used to using currents, indeed had used them in my thesis. When I did the calculations for the weightings on squares and discs and the like, it was the idea of currents that was brought to mind by the pictures.

Posted by: Simon Willerton on April 22, 2010 7:14 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Do you want to expand on this?

Sure. To begin with, to someone who works with measures a lot, Tom’s original definition of a weighting vector ww for a finite metric space XX brings measures to mind immediately. After all, how does ww show up? The weighted sums xXe d(y,x)w x=1\sum_{x\in X} e^{-d(y,x)} w_x=1 can be interpreted as integrals of the functions f y(x)=e d(y,x)f_y(x) = e^{-d(y,x)} and the sum xXw x=|X|\sum_{x\in X} w_x = \vert X \vert is then the total (signed) measure of XX. Even the name “weighting” brings to mind common terminology for discrete measures. If XX is no longer finite, no problem, just forget that we ever thought of ww as a vector: now it’s a signed measure on XX.

But we don’t want to fall into the trap of thinking everything is a nail just because we happen to have a hammer handy. Is there some reason that measures arise naturally from the mathematics here? Well, another way to think about the finite sums above is to identify X\mathbb{R}^X with its dual space and consider w( X) *w \in (\mathbb{R}^X)^*, so what we have is w(f y)=1w(f_y) = 1 for every yXy\in X and then |X|=w(1)\vert X \vert = w(1). When XX is infinite, we would like to replace X\mathbb{R}^X with some function space on XX which contains all the functions f yf_y and 11; this will be an infinite dimensional space, most likely not identifiable with its dual in a reasonable way, and so ww will become a different kind of creature.

So which function space should we work with? One that comes to mind is C b(X)C_b(X), the space of bounded continuous real-valued functions on XX. (Keep in mind we don’t have a lot of structure to work with. If XX is nothing but a metric space with no canonical choice of a measure we can’t talk about L pL^p spaces, for example.) C b(X)C_b(X) contains the functions f yf_y and 11, and is a Banach space with the sup norm. If XX is moreover compact (so we can drop that bb), then C(X) *C(X)^* is isometrically isomorphic to M(X)M(X), the space of signed Borel measures on XX; a signed measure acts on a continuous function by integration. So measures really do show up naturally.

I said some abbreviated version of this to Tom in email. He said that he didn’t find this argument completely satisfactory in part because the space C(X)C(X) only depends on the topology of XX, and doesn’t see anything else about the metric structure. (Tom, please correct me if I’m misrepresenting or leaving out something important.) He suggested that it might make more sense to use Lip(X)Lip(X), the space of Lipschitz functions on XX, or something like it. Lip(X)Lip(X) is also a Banach space with an appropriate norm, which contains f yf_y and 11. Furthermore Lip(X) *Lip(X)^* (is isometrically isomorphic to a space which) contains M(X)M(X), but as a proper subspace when XX is infinite. Heuristically, since Lipschitz functions are within spitting distance of being differentiable (when “differentiable” even makes sense), the elements of Lip(X) *Lip(X)^* are something like functionals of differentiable functions, so they are something like some class of distributions, or currents. (I’d never actually heard of currents until recently, when I was talking to a differential geometer colleague about this.) So maybe that’s the sort of thing ww should be.

I should point out one possible objection to Tom’s hesitation about using C(X)C(X) and M(X)M(X): although those Banach spaces don’t know much about the metric on XX, the functions f yf_y do, and they stay in the picture in any case.

Of course this is not just an aesthetic or philosophical issue. We want definitions of weighting and magnitude that let us prove good theorems. In particular, I’d like a definition of magnitude which is equivalent to yours whenever a weighting measure exists. In the framework I’ve described, this means that whatever function space we work with, we want its dual to contain M(X)M(X). It would also be nice to have a definition such that the definitions of magnitude in your paper with Tom, via approximations by finite sets, become theorems (this is what Tom said to me that set me to thinking about all of the above in the first place). To prove such theorems we need to know something about how measures with finite support sit inside our dual space. For both of the function spaces C(X)C(X) and Lip(X)Lip(X) I do know some interesting things about this question, but I haven’t managed to use them prove the desired theorems yet.

[Off-topic question for those who understand the tex rendering here: I didn’t bother to make the Lip in Lip(X)Lip(X) Roman, although I normally would, out of laziness. But it appears in Roman anyway, on my browser at least. Why is that?]

Posted by: Mark Meckes on April 23, 2010 3:30 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark’s comments here just suggested something which might be of passing interest if not of real relevance. Since I’m currently a bit sleep-deprived and not really up to cogent mathematical thought today, I’d just like to throw out some disjointed remarks and half-remembered tidbits. Apologies if this derails the discussion a bit; I am really just commenting here while the iron is hot.

- The use of lip(X) or Lip(X) as a kind of coordinate ring (broadly speaking) for compact metric spaces has been championed in some papers and in a book of Nik Weaver, this POV probably goes back to work of Mark Rieffel but I forget the precise referencs.

- Lip(X) is the bidual Banach space of lip(X), and the predual of Lip(X) is I think a named widget, called the Arens-Eells space of X – it is like some linear combination of “molecules” or “atoms” - hopefully someone who’s seen some modern-era Hardy space theory can correct or expand on this bit? I don’t know about Lip(X)* though.

- I seem to remember that the forgetful functor from (Banach spaces, linear maps of norm $\leq 1$) to (metric spaces, distance-non-increasing maps) has a left adjoint. I first saw this in a talk of Gilles Godefroy on some work he did with Nigel Kalton on “Lipschitz-free spaces” - although he didn’t once use the words functor or adjunction if I recall correctly.

Posted by: Yemon Choi on April 23, 2010 10:22 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Let me make some general comments as to why Lipschitz functions might be appropriate in this context, or at least let me speculate on why Tom might think so.

The main point is that the idea of magnitude of metric spaces arose from thinking of metric space as enriched categories (following Lawvere). In that situation, the obvious notion of function (though probably not the only one) is that of Lipschitz function. Lipschitz functions play the role of ‘enriched presheaves’. I’ll try to explain what kind of properties can be deduced from that point of view.

Firstly, I should say what I assume you mean by Lipschitz functions, or rather what I would like you to mean. I will use [0,][0,\infty] to denote the extended non-negative reals 0\mathbb{R}_{\ge 0}\cup\infty. In the present context I would think of a Lipschitz function as being a function h:X[0,]h: X\to [0,\infty] such that d(x,y)h(x)h(y)d(x,y)\ge h(x) - h(y). A good class of examples, indeed a key class of examples, of such functions is given in the following way. Take any yXy\in X and define the function h y:X[0,]h_y:X\to [0,\infty] by h y(x)d(x,y)h_y(x)\coloneqq d(x,y). The triangle inequality ensures that this satisfies the Lipschitz condition. More generally, for any subset YXY\subset X we can define h Y:X[0,]h_Y:X\to [0,\infty] by h Y(x)inf yYd(x,y)h_Y(x)\coloneqq inf_{y\in Y} d(x,y). I’ll come back to the relevance of these in a second, but the sharp-eyed category theorists should be shouting “Yoneda!” at their screen by now.

Now this wasn’t quite what you were talking about. You were talking about functions like f y:X[0,1]f_y:X\to [0,1] where f y=e h yf_y=e^{h_y}, so I would expect that your functions f:X[0,1]f:X\to [0,1] will satisfy a ‘multiplicative Lipschitz condition’ f(x)/f(y)e d(x,y)f(x)/f(y)\ge e^{-d(x,y)} for all xx and yy. So you might want to translate what I’m going to say over to your kind of functions.

Now Lip(X)Lip(X) the set of Lipschitz functions (in my sense) carries a ‘generalized metric’ defined on function hh' and hh'' as follows:

d(h,h)sup xX[h(x)h(x)]d(h', h'')\coloneqq sup_{x\in X} [h''(x)- h'(x)]

where my ad-hoc notation [a]max(0,a)[a]\coloneqq max(0, a) is the ‘non-negative part’. This generalized metric is not symmetric, might take the value \infty, and d(h,h)=0d(h',h'')=0 does not imply that h=hh'=h''; however it does satisfy the triangle equality and d(h,h)=0d(h,h)=0.

The key point about the above generalized metric is that the function

XLip(X);yh yX\to Lip(X); \quad y\to h_y

is an isometric embedding – it’s a nice exercise to prove that. [For the cognoscetti this is an enriched version of the Yoneda embedding, Lip(X)Lip(X) is the [0,][0,\infty]-category of [0,][0,\infty]-presheaves on XX.] As John B keeps telling us in a more general setting, we should think of Lip(X)Lip(X) as some kind of cocompletion of XX. (Okay, yes, when people bandy about words like cocompletion it causes some glazed expression to pass across my eyes.)

That the set of Lipschitz functions Lip(X)Lip(X) is a cocompletion means that it will have all colimits (whatever that means), in particular it will have coproducts, so we will be able to add things together. In XX we can’t add two points together, but we can in Lip(X)Lip(X): it makes sense in many circumstances to take the distance from a point xx to the union of two points yy and yy' to be the minimum of distances from xx to yy and from xx to yy'. In this way we can think of the union of two points as being a Lipschitz function.

This leads to thinking of any subset of points in XX as giving an element of Lip(X)Lip(X) as we did above. For YXY\subset X define the function h YLip(X)h_Y\in Lip(X) by h Y(x)inf yYd(x,y)h_Y(x)\coloneqq inf_{y\in Y} d(x,y).

Things get even nicer when we restrict to Comp(X)Comp(X), the set of compact subsets of XX. Firstly, if YY is compact then we can recover YY from the function h Yh_Y by just taking the zero set of the function. Moreover, we can define a generalized metric on Comp(X)Comp(X) by saying d(Y,Y)=ϵd(Y',Y'')=\epsilon if ϵ\epsilon is the least number such that YY' is contained in a closed ϵ\epsilon-neighbourhood of YY''. In particular d(Y,Y)=0d(Y',Y'')=0 if and only if YYY'\subset Y''. We obtain the Haussdorf metric (a genuine metric) DD on Comp(X)Comp(X) by symmetrizing dd in the following way:

D(Y,Y)max(d(Y,Y),d(Y,Y)).D(Y',Y'')\coloneqq max\bigl( d(Y',Y''),d(Y'',Y')\bigr).

Although I guess the Haussdorf metric has less information in it. None-the-less, we get an isometric embedding of generalized metric spaces:

(Comp(X),d)(Lip(X),d);Yh Y.(Comp(X),d)\to (Lip(X),d);\quad Y\to h_Y.

This can be made into an isometric embedding of genuine metric spaces by using the Haussdorf metric on the set of compact subsets and similarly symmetrizing the generalized metric on the set of Lipschitz functions.

So Lip(X)Lip(X) contains not only the points of XX but also the unions of points of XX. I guess there’s lots more stuff in Lip(X)Lip(X), but I’m not imaginative enough at this point in time to think of anything else. What other colimits would be in there other than coproducts?

Anyway, this is all very natural from the point of view of enriched category theory and everything I’ve said has analogues in other enriched situations. As magnitude of metric spaces arose by thinking of metric spaces as being enriched categories, it makes sense to hope/think/guess that these sorts of functions are the right ones to consider in this context.

Posted by: Simon Willerton on April 24, 2010 3:38 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Thanks for the comments, Simon (and thanks also to Yemon - I’ve only recently started learning about Lip(X)Lip(X) and lip(X)lip(X) and all that, so pointers to the literature are good to have [1]).

To me your comments actually support working with C(X)C(X) instead of Lip(X)Lip(X). Let me explain why. First of all, I’m going to work only with symmetric metrics, not because I disagree with Lawvere but because I don’t know how much classical stuff I know will still work without symmetry [2]. For me, a function f:Xf:X\to \mathbb{R} is LL-Lipschitz for L0L \ge 0 if sup x,yX|f(x)f(y)|Lsup_{x,y\in X}|f(x)-f(y)| \le L, so what you call Lipschitz is what I call 11-Lipschitz; ff is said to be Lipschitz if it is LL-Lipschitz for some (finite) LL. The functions f y(x)=e d(x,y)f_y(x)=e^{-d(x,y)} are 11-Lipschitz, since te tt \mapsto e^{-t} is 11-Lipschitz on [0,)[0,\infty).

Since I hope to use Banach space methods here, for me Lip(X)Lip(X) is the set of all Lipschitz functions f:XRf:X\to \R, since 11-Lipschitz functions alone don’t form a vector space. Next I need a norm. The symmetric analogue to the distance you put on 11-Lipschitz functions is the sup norm: f =sup xX|f(x)|\|f\|_\infty = \sup_{x\in X} |f(x)|. Now Lip(X)Lip(X) is not complete in this norm. It’s a dense subspace of C(X)C(X), and so (Lip(X), ) *=M(X)(Lip(X), \| \cdot \|_\infty)^* = M(X). So your comments seem to support what I meant by “working with C(X)C(X)”. By “working with Lip(X)Lip(X)” I mean instead equipping Lip(X)Lip(X) with a different norm (the bounded Lipschitz norm) which makes it a Banach space, whose dual (and in fact predual) contain M(X)M(X) as a proper subspace.

[1] One thing I will point out related to Yemon’s comments is that finitely supported signed measures are norm-dense in the predual of Lip(X)Lip(X) (with the bounded Lipschitz norm). This is one thing I think could be useful in this context. Note that finitely supported signed measures are not norm-dense in M(X)M(X), although they are dense in the weak-* topology.

[2] In case you ever meet a skeptical analyst or geometer who might think Lawvere lacks the metric-space-credibility to advocate dropping such a fundamental axiom, you can tell them that Gromov has also argued against the symmetry axiom as too restrictive for applications.

Posted by: Mark Meckes on April 24, 2010 6:48 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

To answer your parenthetical question about typesetting. In itex (which is what is used here) consecutive, abutting Roman letters in mathmode are interpreted as being a word, so are typeset upright.

  • $L i p$ gives LipL i p
  • $Lip$ gives LipLip
Posted by: Simon Willerton on April 24, 2010 4:00 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

On page 3, you write “It does seem that the subspace distance metric and the intrinsic metric are infinitessimally the same. I would hope that the asymptotics of the magnitude only depend on metric infinitessimally.” I though I found a counter-example to this here (and you did as well, I think?)

I would not say that I hoped something was true if I knew a counter-example.

Also, on page 6, “we will note distinguish” should read “we will not distinguish”.

Posted by: David Speyer on April 22, 2010 3:37 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I’m glad you’re there to keep me on my toes David.

I guess I had in my memory that there was some problem with your calculation for the subspace metric on S 2S^2 (I don’t think I could quite figure it out or something, I don’t remember). But I hadn’t got round to going back to look at it. Anyway, I just sat down and calculated the thing on the nose today. It’s very nice. As you said, the thing to calculate is

2π θ=0 πe R2sin(θ/2)sin(θ)dθ.2\pi\int^\pi_{\theta=0} e^{-R 2\sin(\theta/2)} \sin(\theta) d \theta.

The cunning observation is that

sin(θ)dθ=2sin(θ/2)cos(θ/2)dθ=2sin(θ/2)d(2sin(θ/2)).\sin(\theta) d \theta = 2\sin(\theta/2)\cos(\theta/2)d\theta =2\sin(\theta/2)d(2\sin(\theta/2)).

So substituting s=2sin(θ/2)s=2\sin(\theta/2) we get that the integral equals 2π s=0 2e Rssds2\pi\int_{s=0}^2 e^{-R s} s d s which evaluates to

2π(1e 2R(2R+1))R 2.\frac{2\pi(1-e^{-2R}(2R+1))}{R^2}.

The magnitude of the radius RR sphere with the subspace metric is the volume of S 2S^2 divided by that integral, whence

|S R,sub 2|=2R 21e 2R(2R+1)=2R 2+exponentially decaying terms.\left|S^2_{R,sub}\right| =\frac{2R^2}{1-e^{-2R}(2R+1)}= 2R^2+exponentially decaying terms.

However, for the intrinsic metric you we have the magnitude is of the form 2R 2+2+O(R 2)2R^2+2 +O(R^{-2}). So asymptotically we get different answers for the magnitude.

You can do a similar calculation for higher dimensional spheres.

I’m sure I only put those questions there about the subspace metric so that you would goad me into actually doing the calculations!

This then leads to the question: in what sense are the subspace metric and the intrinsic metric infinitesimally the same? I still haven’t got my head round this.

Posted by: Simon Willerton on April 23, 2010 7:10 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

in what sense are the subspace metric and the intrinsic metric infinitesimally the same?

One sense in which this is true is that an ε\varepsilon-ball w.r.t. the intrinsic metric is the same as a 2Rsin(ε/2R)2R \sin (\varepsilon / 2R)-ball w.r.t. the subspace metric, and 2Rsin(ε/2R)ε2R\sin (\varepsilon / 2R) \approx \varepsilon as ε0\varepsilon \to 0.

Posted by: Mark Meckes on April 23, 2010 12:34 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

You say you haven’t calculated the magnitudes of spheres with the subspace metric yet. If you haven’t even checked for S 1S^1 yet, I just did and it’s easy to see (the hardest part is using a trig identity) that it matches the expression derived in your paper with Tom.

Also, a typo in Theorem 1: μ\mu is an invariant measure.

Posted by: Mark Meckes on April 22, 2010 3:43 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Oh, I see the calculation for S 1S^1 is also in the discussion David just linked to.
Posted by: Mark Meckes on April 22, 2010 3:48 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Mark said

You say you haven’t calculated the magnitudes of spheres with the subspace metric yet. If you haven’t even checked for S 1S^1 yet, I just did and it’s easy to see (the hardest part is using a trig identity) that it matches the expression derived in your paper with Tom.

I’m not sure which bit you mean you’ve checked!

I should say certainly say a little more in the paper about S 1S^1, as we do know a reasonable amount about it. Of course what I meant to say was “I haven’t calculated it for the general nn-sphere, but we know a fair bit about the circle, though I don’t really want to generalize from that.” However, that isn’t what I said. It was one of the bits of the paper where I needed to fill in the blank before finishing it and I rushed it. Thanks.

In the paper with Tom, it was shown that that the magnitude of a circle has the expected asymptotic behaviour – namely half the length – when it is thought of a subspace of any constant curvature surface. In particular this covers both the Euclidean metric and the intrinsic metric. The non-asymptotic magnitude will in general be different metrics on the circle.

Posted by: Simon Willerton on April 22, 2010 7:05 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

What I checked is that the magnitude of a circle of arbitary fixed circumference, as a subspace of 2\mathbb{R}^2, has the same magnitude whether defined as in the present paper, or as in section 4.1 of your paper with Tom.

Actually I think this has to be the case from measure-theoretic “first principles”, but there are some fiddly functional-analytic issues I want to think about more before I say I’m sure of that.

Posted by: Mark Meckes on April 22, 2010 7:14 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

I tried to post another comment on this earlier but it vanished. In fact the reason they have to be the same appeared in your paper with Tom: the sums for the finite approximations of the circle are just Riemann sums for the integral corresponding to the whole circle.

To make it sound fancier, the measures μ n\mu_n, which give mass 1/n1/n to each of nn equally spaced points on a circle, converge in the weak-* topology of M(S 1)M(S^1) (induced by the action as the dual of C(S 1)C(S^1)) to the invariant probability measure on S 1S^1.

Posted by: Mark Meckes on April 23, 2010 12:58 AM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

It seems I should have waited a while to collect my comments before posting, but it’s too late for that now.

Regarding Theorem 1 (Speyer’s Homogeneous Magnitude Theorem), it’s worth pointing out that if XX is a compact homogeneous metric space then (up to normalization) there exists a unique invariant measure μ\mu on XX.

Posted by: Mark Meckes on April 22, 2010 4:30 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Furthermore this is a positive measure, so xXe d(x,y)dμ\int_{x\in X} e^{-d(x,y)} d\mu is automatically nonzero and so the magnitude of XX is indeed defined.

Folklore has it that a nonzero invariant measure on a noncompact space is (typically at least) infinite. Does anyone know an actual theorem to that effect?

Posted by: Mark Meckes on April 22, 2010 7:04 PM | Permalink | Reply to this

Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces

Is http://arxiv.org/abs/math-ph/0210033 “Volumes of Compact Manifolds” of some use, here?

Posted by: Alejandro Rivero on April 27, 2010 1:57 AM | Permalink | Reply to this
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