## January 10, 2018

### On the Magnitude Function of Domains in Euclidean Space, I

#### Posted by Simon Willerton

guest post by Heiko Gimperlein and Magnus Goffeng.

The magnitude of a metric space was born, nearly ten years ago, on this blog, although it went by the name of cardinality back then. There has been much development since (for instance, see Tom Leinster and Mark Meckes’ survey of what was known in 2016). Basic geometric questions about magnitude, however, remain open even for compact subsets of $\mathbb{R}^n$: Tom Leinster and Simon Willerton suggested that magnitude could be computed from intrinsic volumes, and the algebraic origin of magnitude created hopes for an inclusion-exclusion principle.

In this sequence of three posts we would like to discuss our recent article, which is about asymptotic geometric content in the magnitude function and also how it relates to scattering theory.

For “nice” compact domains in $\mathbb{R}^n$ we prove an asymptotic variant of Leinster and Willerton’s conjecture, as well as an asymptotic inclusion-exclusion principle. Starting from ideas by Juan Antonio Barceló and Tony Carbery, our approach connects the magnitude function with ideas from spectral geometry, heat kernels and the Atiyah-Singer index theorem.

We will also address the location of the poles in the complex plane of the magnitude function. For example, here is a plot of the poles and zeros of the magnitude function of the $21$-dimensional ball.

We thank Simon for inviting us to write this post and also for his paper on the magnitude of odd balls as the computations in it rescued us from some tedious combinatorics.

The plan for the three café posts is as follows:

1. State the recent results on the asymptotic behaviour as a metric space is scaled up and on the meromorphic extension of the magnitude function.

2. Discuss the proof in the toy case of a compact domain $X\subseteq \mathbb{R}$ and indicate how it generalizes to arbitrary odd dimension.

3. Consider the relationship of the methods to geometric analysis and potential ramifications; also state some open problems that could be interesting.

### Asymptotic results

As you may recall, the magnitude $\mathrm{mag}(X)$ of a finite subset $X\subseteq \mathbb{R}^n$ is easy to define: let $w:X\to \mathbb{R}$ be a function which satisfies

$\sum_{y\in X} \mathrm{e}^{-\mathrm{d}(x,y)} w(y) = 1 \quad \text{for all}\ x \in X.$

Such a function is called a weighting. Then the magnitude is defined as the sum of the weights:

$\mathrm{mag}(X) = \sum_{x \in X} w(x).$

For a compact subset $X$ of $\mathbb{R}^n$, Mark Meckes shows that all reasonable definitions of magnitude are equal to what you get by taking the supremum of the magnitudes of all finite subsets of $X$:

$\mathrm{mag}(X) = \sup \{\mathrm{mag}(\Xi) : \Xi \subset X \ \text{finite} \} .$

Unfortunately, few explicit computations of the magnitude for a compact subset of $\mathbb{R}^n$ are known.

Instead of the magnitude of an individual set $X$, it proves fruitful to study the magnitude of dilates $R\cdot X$ of $X$, for $R\gt 0$. Here the dilate $R\cdot X$ means the space $X$ with the metric scaled by a factor of $R$. We can vary $R$ and this gives rise to the magnitude function of $X$:

$\mathcal{M}_X\colon (0,\infty)\to \mathbb{R};\quad\mathcal{M}_X(R) := \mathrm{mag}(R\cdot X)\quad\text{for }\ R \gt 0.$

Tom and Simon conjectured a relation between the magnitude function of $X$ and its intrinsic volumes $(V_i(X))_{i=0}^n$. The intrinsic volumes of subsets of $\mathbb{R}^n$ generalize notions such as volume, surface area and Euler characteristic, with $V_n(X)=\text{vol}_n(X)$ and $V_0(X)=\chi(X)$.

Convex Magnitude Conjecture. Suppose $X \subseteq \mathbb{R}^n$ is compact and convex, then the magnitude function is a polynomial whose coefficients involve the intrinsic volumes:
$\mathcal{M}_X(R) = \sum_{i=0}^n \frac{V_i(X)}{i! \,\omega_i} R^n,$ where $V_i(X)$ is the $i$-th intrinsic volume of $X$ and $\omega_i$ the volume of the $i$-dimensional ball.

The conjecture was disproved by Barceló and Carbery (see also this post). They computed the magnitude function of the $5$-dimensional ball $B_5$ to be the rational function $\mathcal{M}_{B_5}(R)=\frac{R^5}{5!} +\frac{3R^5+27R^4+105R^3+216R+72}{24(R+3)}.$ In particular, the magnitude function is not even a polynomial for $B_5$. Also, the coefficient of $R^4$ in the asymptotic expansion of $\mathcal{M}_{B_5}(R)$ as $R \to \infty$ does not agree with the conjecture.

Nevertheless, for any smooth, compact domain in odd-dimensional Euclidean space, $X\subseteq\mathbb{R}^n$ (in other words, the closure of an open bounded subset with smooth boundary), for $n=2m-1$, our main result shows that a modified form of the conjecture holds asymptotically as $R \to \infty$.

Theorem A. Suppose $n=2m-1$ and that $X\subseteq \mathbb{R}^n$ is a smooth domain.

1. There exists an asymptotic expansion of the magnitude function: $\mathcal{M}_X(R) \sim \frac{1}{n!\omega_n}\sum_{j=0}^\infty c_j(X) R^{n-j} \quad \text{as }\ R\to \infty$ with coefficients $c_j(X)\in\mathbb{R}$ for $j=0,1,2,\ldots$.

2. The first three coefficients are given by \begin{aligned} c_0(X)&=\text{vol}_n(X),\\ c_1(X)&=m\text{vol}_{n-1}(\partial X),\\ c_2(X)&=\frac{m^2}{2} (n-1)\int_{\partial X} H \ \mathrm{d}S, \end{aligned} where $H$ is the mean curvature of $\partial X$. (The coefficient $c_0$ was computed by Barceló and Carbery.) If $X$ is convex, these coefficients are proportional to the intrinsic volumes $V_{n}(X)$, $V_{n-1}(X)$ and $V_{n-2}(X)$ respectively.

3. For $j\geq 1$, the coefficient $c_j(X)$ is determined by the second fundamental form $L$ and covariant derivative $\nabla_{\partial X}$ of $\partial X$: $c_j(X)$ is an integral over $\partial X$ of a universal polynomial in the entries of $\nabla_{\partial X}^k L$, $0 \leq k\leq j-2$. The total number of covariant derivatives appearing in each term of the polynomial is $j-2$.

4. The previous part implies an asymptotic inclusion-exclusion principle: if $A$, $B$ and $A \cap B \subset \mathbb{R}^n$ are smooth, compact domains, $\mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as }\ R \to \infty$ faster than $R^{-N}$ for all $N$.

If you’re not familiar with the second fundamental form, you should think of it as the container for curvature information of the boundary relative to the ambient Euclidean space. Since Euclidean space is flat, any curvature invariant of the boundary (satisfying reasonable symmetry conditions) will only depend on the second fundamental form and its derivatives.

Note that part 4 of the theorem does not imply that an asymptotic inclusion-exclusion principle holds for all $A$ and $B$, even if $A$ and $B$ are smooth, since the intersection $A \cap B$ usually is not smooth. In fact, it seems unlikely that an asymptotic inclusion-exclusion principle holds for general $A$ and $B$ without imposing curvature conditions, for example by means of assuming convexity of $A$ and $B$.

The key idea of the short proof relates the computation of the magnitude to classical techniques from geometric and semiclassical analysis, applied to a reformulated problem already studied by Meckes and by Barceló and Carbery. Meckes proved that the magnitude can be computed from the solution to a partial differential equation in the exterior domain $\mathbb{R}^n\setminus X$ with prescribed values in $X$. A careful analysis by Barceló and Carbery refined Meckes’ results and expressed the magnitude by means of the solution to a boundary value problem. We refer to this boundary value problem as the “Barceló-Carbery boundary value problem” below.

### Meromorphic extension of the magnitude function

Intriguingly, we find that the magnitude function $\mathcal{M}_X$ extends meromorphically to complex values of the scaling factor $R$. The meromorphic extension was noted by Tom for finite metric spaces and was observed in all previously known examples.

Theorem B. Suppose $n=2m-1$ and that $X\in \mathbb{R}^n$ is a smooth domain.

• The magnitude function $\mathcal{M}_X$ admits a meromorphic continuation to the complex plane.

• The poles of $\mathcal{M}_X$ are generalized scattering resonances, and each sector $\{z : |\arg(z)| \lt \theta \}$ with $\theta \lt \frac{\pi}{2}$ contains at most finitely many of them (all of them outside $\{z : |\arg(z)|\lt {\textstyle \frac{\pi}{n+1}}\}$).

The ordinary notion of scattering resonances comes from studying waves scattering at a compact obstacle $X\subseteq \mathbb{R}^n$. A scattering resonance is a pole of the meromorphic extension of the solution operator $(R^2-\Delta)^{-1}$ to the Helmholtz equation on $\mathbb{R}^n\setminus X$, with suitable boundary conditions. These resonances determine the long-time behaviour of solutions to the wave equation and are well studied in geometric analysis as well as throughout physics. The Barceló-Carbery boundary value problem is a higher order version of this problem and studies solutions to $(R^2-\Delta)^{m}u=0$ outside $X$. In dimension $n=1$ (i.e. $m=1$), the Barceló-Carbery problem coincides with the Helmholtz problem, and the poles of the magnitude function are indeed scattering resonances. As in scattering theory, one might hope to find detailed structure in the location of the poles. A plot of the poles and zeros in case of the $21$-dimensional ball is given at the top of the post.

The second part of this theorem is sharp. In fact, the poles do not need to lie in any half plane. Using the techniques of Barceló and Carbery we observe that the magnitude function of the 3-dimensional spherical shell $X=(2B_3)\setminus B_3^\circ$ is not rational and contains an infinite discrete sequence of poles which approaches the curve $\mathrm{Re}(R)= \log(|\mathrm{Im}(R)|)$ as $\mathrm{Re}(R) \to \infty$. Here’s a plot of the poles of $\mathcal{M}_{X}$ with $|\mathrm{Im}(R)|\lt 30$.

The magnitude function of $X$ is given by $\mathcal{M}_X(R)=(7/6)R^3 +5R^2 + 2R + 2 + \frac{e^{-2R}(R^2+1)+2R^3-3R^2+2R-1}{\sinh 2R -2R}.$

[EDIT: The above formula has been corrected, following comments below.]

Our techniques extend to compact domains with a $C^k$ boundary, as long as $k$ is large enough. In this case, the asymptotic inclusion-exclusion principle takes the form that $\mathcal{M}_{A \cup B}(R) - \mathcal{M}_A(R) - \mathcal{M}_B(R) + \mathcal{M}_{A \cap B}(R) \to 0 \quad \text{as}\ R \to \infty$ faster than $R^{-N}$ for an $N=N(k)$.

Posted at January 10, 2018 5:11 PM UTC

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### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Very interesting stuff! The question that leaps out to me is “Do you know anything about $c_3(X)$?” The obvious guess would be something proportional to the total scalar curvature of $\partial X$, which is $V_{n-3}(X)$ for convex $X$. However, if memory serves me right, that is the integral of the second symmetric polynomial in the eigenvalues of the second fundamental form, so doesn’t involve the derivative of the second fundamental form, so your theorem seems to be indicating that it possibly isn’t that.

Posted by: Simon Willerton on January 10, 2018 5:52 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Thanks Simon! And thanks for the question! Our results come with a method to compute all the numbers $c_j(X)$. For $j\geq 2$ the computations are quite hairy so we were happy to go as far as $c_2(X)$. There are related computations of far many more coefficients in the short time asymptotics of the heat kernel on a compact manifold by Gilkey. The description of the coefficients $c_j(X)$ as an “an integral over $\partial X$ of a universal polynomial ” in covariant derivatives of the second fundamental form of order $k\leq j-2$ is the general scenario. There are (to my limited knowledge) only three potential terms that could enter: the square of the mean curvature, the second order mean curvature and the scalar curvature. It is currently unclear how much one could simplify the expression for $c_3(X)$, perhaps it is only a scalar curvature term that enters.

Similar computations have been done in a closely related context by Sher-Polterovich for the short time asymptotics of the heat kernel of the Dirichlet-to-Neumann operator (the DN-operator for the usual Laplacian, not our the higher order problem relevant for magnitudes) and the relevant term (see Formula (1.4.4) on page 3 of https://arxiv.org/abs/1304.7233) contains all three possible curvature terms.

The idea we applied in computing $c_2(X)$ was to show that it was proportional to the integral of the mean curvature of the boundary then to deduce the value of the proportionality constant from your oddball computations. Since one a priori knew that there are few curvature terms appearing in $c_3(X)$, one silly approach could be to compute 3 classes of examples and solve a linear equation in the 3 unkown proportionality constants (that will only depend on the dimension).

Posted by: Magnus Goffeng on January 11, 2018 9:13 AM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

I have another question. For the three-dimensional shell, do you calculate the potential function? Is it just some linear combination of $\psi_0(r)=e^{-r}$ and $\psi_1(r)=e^{-r}/r$ on the outside of the shell and of $\psi_0(r)+\psi_0(-r)=2\cosh(r)$ and $\psi_1(r)+\psi_1(-r)=-2\sinh(r)/r$ on the inside of the shell?

Posted by: Simon Willerton on January 10, 2018 7:47 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Thank you for your interest, Simon! Indeed, we calculate the potential function in a similar way to Barcelo-Carbery’s or your paper, by solving the PDE in radial coordinates. As you say, in the outside the potential function is the same as for the 3-ball, in the interior it is a messy linear combination of $e^{\pm Rr}$ and $(e^{Rr}-e^{-Rr})/r$.

While the magnitude function is not very difficult to calculate in this example, we found the result instructive: It is no longer a rational function and has an infinite number of poles, which one can write down almost exactly. In particular, there exist poles of arbitrarily large real part and they are “logarithmically close” to the real axis.

Posted by: Heiko Gimperlein on January 10, 2018 10:04 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

I get a different answer to you for the magnitude function of the shell.

Firstly, is there a typo in your leading order term? This ought to be $(7/6)R^3$, oughtn’t it? The volume of the shell $X$ is $(2^3 - 1^3)\omega_3= 7\omega_3$ and the leading order term should be $vol(X)R^3/3!\,\omega_3=(7/6) R^3$. Right?

Secondly, I get a subtlely different expression to you. We can rewrite your expression as polynomial plus exponentially decaying part.

$(7/3)R^3 + 5R^2 +R +4 + \frac{e^{-2R}(R^2+3)+2R^3-7R^2+6R-3}{\sinh 2R -2R}$

Here the quadratic term is correct as the area of the boundary is $5R^2\sigma_2$ where $\sigma_2$ is the area of the unit sphere.

The answer that I get is the following, which is strangely close to yours.

$(7/6)R^3 +5R^2 + 2R + 2 + \frac{e^{-2R}(R^2+1)+2R^3-3R^2+2R-1}{\sinh 2R -2R}$

I have to say that I like my answer better because the constant term is the Euler characteristic of the shell :-)

Both of them have limit of $1$ as $R\to 0$, I think.

Have you got the explicit formula for the potential function inside the shell?

Posted by: Simon Willerton on January 11, 2018 1:37 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Dear Simon,

Thank you very much for your careful reading! It is a bit embarrassing, but we indeed seem to have lost a term here.

I think I agree with your corrected formula, and we are going to fix it in the paper. Because the poles were correct, all the conclusions we draw from this example remain valid.

Posted by: Heiko Gimperlein on January 11, 2018 11:22 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

I’ve updated the formula in the post.

Posted by: Simon Willerton on January 12, 2018 9:55 AM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Let’s think about the coefficient of $R$. As $m=2$ and $n=2m-1=3$, according to your theorem this should be

$\frac{1}{3!\,\omega_3}\frac{2^2}{2}2\int_{\partial X} H \mathrm{d}S=\frac{2}{3\,\omega_3}\left(\int_{\partial_{inside} X} H \mathrm{d}S + \int_{\partial_{outside} X} H \mathrm{d}S\right).$

Assuming that the mean curvature is going to have opposite sign on the inside and outside boundaries (I guess you want outward pointing normal or something) this gives

$\frac{2}{3\,\omega_3}\left(-\int_{S^2} 1 \,\mathrm{d}S + \int_{2S^2} \frac{1}{2}\, \mathrm{d}S\right)=\frac{2\sigma_2}{3\,\omega_3}\left(-1 + 2\right)=2.$

(In general you have $\sigma_{2p}=(2p+1)\omega_{2p+1}$.)

So the coefficient of $R$ should be $2$, and not $1$ as it seems to be in your result.

Posted by: Simon Willerton on January 11, 2018 2:29 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Now it seems that we’ve settled on the magnitude function of the spherical shell, let me return to my comment about preferring my answer because it contains the Euler characteristic.

We have for $X$ the spherical shell

$\mathcal{M}_X(R)=\frac{7}{6}R^3 +5R^2 + 2R + 2 + \frac{e^{-2R}(R^2+1)+2R^3-3R^2+2R-1}{\sinh 2R -2R},$

and for $B^3$ the unit three-ball we know from Barceló and Carbery’s calculations that

$\mathcal{M}_{B^3}(R)=\frac{1}{6}R^3 + R^2 + 2 R +1$

In the asymptotic, polynomial part we have geometric interpretation of the first three coefficients $c_0$, $c_1$ and $c_2$. We know that for a $3$-dimensional domain $M^3$ that the constant term $c_3(M^3)$ is some kind of curvature measure of the boundary and might well be proportional to $\int_{\partial M^3} K\, \mathrm{d}S$, where $K$ is the Gauss curvature of the boundary. By the Gauss-Bonnet Theorem this is proportional to Euler characteristic of the boundary, or equivalently(*), proportional to the Euler charactistic of the domain $M^3$ itself.

Let me now just observe that $\chi(X)=2$ and $\chi(B^3)=1$ and note that these match with the constant terms… Of course, as Magnus points out above, we would need some more data to pin this down properly. But this looks intriguing.

(*) An under-appreciated fact is that for $M$ an odd-dimensional manifold with boundary we have $\chi(M) = \chi(\partial M)/2$. As Schanuel points out in his “What is the length of a potato?paper, if you have a knotted pile of string and want to know how many pieces of string there are, any child would be able to tell you that you should count the number of end points and divide by 2!

Posted by: Simon Willerton on January 12, 2018 9:50 AM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Dear Simon,

Indeed, that the constant term $c_n$ is the Euler characteristic in all known examples is intriguing.

One might be able to check this in small dimensions by calculating $c_n$, and we have some preliminary ideas that one might prove it for closed surfaces.

This question is also one of the open problems we are going to mention in our third post, and (as explained there) is reminiscent of the Atiyah-Singer index theorem.

Posted by: Heiko Gimperlein on January 12, 2018 4:37 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Thanks for this post!

Can you say anything more about the regularity of $\mathcal{M}_X$ when $X$ is convex? Does it have only finitely many poles in that case? Is it rational?

Posted by: Mark Meckes on January 11, 2018 3:00 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

Dear Mark,

Thank you. This is a question we have asked ourselves, but we do not have a good answer yet. In the paper our focus was on general domains $X$, convex or not.

The magnitude function of a convex $X$ is better behaved in some ways: For example, for convex Euclidean domains the $c_0, c_1, c_2$ depend continuously on $X$ in the Gromov-Hausdorff topology (because they are intrinsic volumes, according to our main theorem).

Whether I would expect the number of poles to be finite for a convex domain, I am not sure. Our current understanding does not indicate this.

Posted by: Heiko Gimperlein on January 11, 2018 6:57 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, I.

I have nothing intelligent to contribute, but thanks, Heiko and Magnus, for this post!

Posted by: Tom Leinster on January 12, 2018 3:20 PM | Permalink | Reply to this

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