## January 23, 2019

### On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective (a)

#### Posted by Simon Willerton

guest post by Heiko Gimperlein and Magnus Goffeng

In this third in a series of blog posts on the magnitude function of Euclidean domains, we shall state from a more general perspective and discuss some new problems that we think might lead to a better understanding of magnitude. The starting point for these problems is in the analogy that our work draws between magnitudes and geometric analysis, an analogy we briefly explained in our first and second blogposts.

While working on magnitudes several questions arose that we have not been able to answer (yet). As they might lead to interesting connections between magnitudes and other fields of mathematics, we hope that they could be of interest to the readers of this blog. The questions come in three different flavors: topological continuity properties, geometric content and (particularly non-asymptotic) analytic results. We list some prototypical problems:

• Topological: In what sense (i.e. in which topology) is the magnitude function continuous in the Euclidean domain?
• Geometric: Can one “magnitude the shape of a drum”?
• Analytic: Is convexity detected by the poles of the magnitude function?

Below you find out more about topological problems, examples and counterexamples. A forthcoming post looks into the geometry and analysis of magnitude.

The magnitude is at first defined as a metric invariant. However, for geometric situations there are geometric tools. The Barceló-Carbery boundary value problem discussed in our previous two posts is one such tool, which allow us to apply the heavy toolbox of geometric analysis (and by extension: complex analysis) to study the magnitude function of domains in $\mathbb{R}^n$. The interplay between the analysis of boundary value problems and the topological and geometric invariants coming out of magnitude relates to classical ideas in geometric analysis and Atiyah-Singer index theorems. The problems we pose are influenced by Leinster-Willerton’s convex magnitude conjecture, but draw heavily on the analogy to geometric analysis.

## Some topological problems

A key open question about the magnitude function is its dependence on the domain: In which topologies is $X \mapsto \mathcal{M}_X$ continuous?

Here is a negative result, motivated by a question of Richard Hepworth. For $\epsilon\in [0,1]$ consider $X_\epsilon:=\{x\in \mathbb{R}^3: 1-\epsilon\leq |x|\leq 1+\epsilon\}.$ So, for $\epsilon=0$, we have the unit sphere and for $\epsilon=1$ we have the ball of radius $2$. The family $(X_\epsilon)_{\epsilon\in [0,1]}$ is continuous in the Hausdorff distance. One can compute that

$\mathcal{M}_{X_\epsilon}(R)=\begin{cases} \frac{2R^2+2}{1-\mathrm{e}^{-\pi R}}, &\epsilon=0,\\ \frac{2\epsilon^2+6\epsilon}{3!}R^3+(2\epsilon^2+2)R^2+4\epsilon R+2+\quad&\\ \quad +\frac{\mathrm{e}^{-R(1-\epsilon)}(R^2(1-\epsilon)^2+1)+2R^3(1-\epsilon)^3-3R^2(1-\epsilon)^2+2R(1-\epsilon)-1}{\sinh(2R(1-\epsilon))-2R(1-\epsilon)}, \;&\epsilon\in (0,1)\\ \frac{8}{3!}R^3+4R^2+4R+1, &\epsilon=1. \end{cases}$

Those who read the first blog post in our series may recognize the expression for $\epsilon =\frac{1}{2}$, and the computations for $\epsilon \in (0,1)$ are more or less the same (after changing coordinates from $[0,1-\epsilon]$ to $[0,1]$). The computation for the unit sphere, $\epsilon=0$, goes back to Willerton. Note that

$\lim_{\epsilon\to 1}\mathcal{M}_{X_\epsilon}(R)=\mathcal{M}_{X_1}(R)\ ,$ $\lim_{\epsilon\to 0}\mathcal{M}_{X_\epsilon}(R)=2R^2+2=\mathcal{M}_{X_0}(R) +O(R^{-\infty}).$

In the limit $\epsilon\to 1$, one observes continuity as might be expected in analogy with a result by Leinster-Meckes for convex domains. Note that the $R^0$ term is the Euler characteristic $\chi(X_\epsilon)$, even when the topology changes. For $\epsilon\to 0$ there is a jump of dimension, and it is no surprise that the pointwise values also have a jump. So the magnitude function is not continuous with respect to the Hausdorff distance, but the asymptotics is the same as $R\to \infty$!

While we were preparing this blog post, Simon Willerton informed us that he had independently carried out the same computation with an alternative method and had come to the same conclusion about the discontinuity as $\epsilon\to 0$.

The magnitude can be calculated from the solution of a differential equation, and for elliptic boundary problems with a second-order operator notions like Mosco and $\Gamma$ convergence have been studied. This may be of interest for general domains, however, as pointed out to us by Richard Hepworth, continuity for very special families of domains may be of most interest for applications. Nina Otter’s work on the relation between magnitude and persistent homology motivates the question:

Problem A. Let $x_1, \dots,x_k \in \mathbb{R}^n$ and for $\epsilon \in [0,\infty)$ set $X_\epsilon = \bigcup_{j=1}^k B(x_j,\epsilon)$, the union of $\epsilon$-balls around these points. Is $\epsilon \mapsto \mathcal{M}_{X_\epsilon}(R)$ continuous?

From Simon Willerton’s formulas for the ball, the function is even real analytic for $k=1$ and $n$ odd. Our own work similarly shows real analyticity in the boring case of very small $\epsilon$ > $0$, any $k$, as long as the balls don’t intersect.

The original magnitude conjecture of Leinster-Willerton contained several far-reaching statements for the magnitude function of a (poly-)convex compact subset of Euclidean space, and one may hope for better continuity properties in this setting. An important general result of Leinster-Meckes says that the magnitude $X\mapsto \mathrm{Mag}(X)$ depends continuously on compact convex domains $X$. So when restricting to convex compact domains, and considering the pointwise values of the magnitude function, we do have a continuous dependence on the set $X$.

If the magnitude function admits an asymptotic expansion, $\mathcal{M}_X(R)\sim \sum_{j=0}^\infty c_j(X)R^{n-j}$, one may interpret the Leinster-Willerton conjecture as a statement about the coefficients $c_j(X)$, $j=0,1,\ldots$. Our methods prove the existence of this expansion for domains with smooth boundaries when $n$ is odd. Let’s denote the space of (possibly non-smooth) compact convex subsets of $\mathbb{R}^n$ by $CoCo(\mathbb{R}^n)$. We let $CoCo_0(\mathbb{R}^n)$ denote the subset of compact convex domains and $CoCo_0^\infty(\mathbb{R}^n)\subseteq CoCo_0(\mathbb{R}^n)$ the subset of compact convex domains with smooth boundary. The space $CoCo(\mathbb{R}^n)$ is a compact Hausdorff space in the topology defined from the Hausdorff metric, and all the inclusions

$CoCo_0^\infty(\mathbb{R}^n)\subseteq CoCo_0(\mathbb{R}^n)\subseteq CoCo(\mathbb{R}^n)$

are dense.

Problem B. For which values of $j$ does the coefficients $c_j(X)$ in the asymptotic expansion $\mathcal{M}_X(R)\sim \sum_{j=0}^\infty c_j(X)R^{n-j}$ depend continuously on $X \in CoCo_0^\infty(\mathbb{R}^n)$ in the Hausdorff metric?

Recall that we only have results for $n$ odd, so we restrict to this case. We note that if $c_j(X)$ depends continuously on $X\in CoCo_0^\infty(\mathbb{R}^n)$ we can extend $c_j$ by continuity to $CoCo(\mathbb{R}^n)$. Additionally, if this holds for $c_j$, a continuity argument shows the asymptotic inclusion/exclusion principle:

$c_j(A)+c_j(B)-c_j(A\cap B)=c_j(A\cup B),$

for $A,B\in CoCo(\mathbb{R}^n)$ with $A\cup B$ convex. In particular, $c_j:CoCo(\mathbb{R}^n)\to \mathbb{R}$ is a Hausdorff-continuous function which is rotation-invariant and satisfies inclusion/exclusion. By Hadwiger’s theorem $c_j$ is then given by intrinsic volumes, and by the scaling property $c_j(tX)=t^{n-j}c_j(X)$ there are constants $\alpha_j$, such that

$c_j(X)=\begin{cases} \alpha_j V_{n-j}(X),\quad &j=0,\ldots, n,\\ 0,& j\gt n. \end{cases}.$

As such, Problem B is closely related to the Leinster-Willerton conjecture and asks for which $j$:s is $c_j(X)$ proportional to an intrinsic volume?

For $j=0,1,2$, $c_j(X)$ depends continuously on $X\in CoCo_0^\infty(\mathbb{R}^n)$ because according to our preprint $c_j(X)$ is proportional to $V_{n-j}(X)$ in this range of $j$’s. Computations of Barceló-Carbery, and later generalizations of Willerton, show that in the case of the unit ball in $\mathbb{R}^n$, $c_j\neq 0$ for most (or all?) $j\gt n$. For these $j$ Problem B therefore has a negative solution.

For $j=3$, Problem B fails due to unpublished computations of Goffeng showing that $c_3(X)$ is proportional to $\int_{\partial X} H^2\mathrm{d}S$, the integral of the square of the mean curvature. $c_3(X)$ is neither an intrinsic volume nor continuous in the Hausdorff metric. While it coincides with the Euler characteristic in the special case of the Euclidean ball in dimension $3$, this is not true for general convex domains $X\subset \mathbb{R}^3$.

The case $j=n$ is of particular interest as it would relate the asymptotic expansion of the magnitude to the Euler characteristic $\chi(X)$. Still, in recent unpublished work, we find that for a closed Riemannian surface $X$ with a metric defined from the geodesic distance, the following asymptotics holds: $\mathcal{M}_X(R)=\frac{\mathrm{vol}(X)}{4\pi}R^2+\chi(X)+O(R^{-2}).$ The asymptotics fits with calculations of Willerton for homogeneous surfaces.

For the family of spaces $(X_\epsilon)_{\epsilon\in [0,1]}$ exhibited above, the coefficient $c_n(X_\epsilon)$ is $2$ for $\epsilon\in [0,1)$ and $c_n(X_1)=1$. So Problem B could not have a positive solution for $j=n$ for all domains.

Problem C. For which $N$’s does it hold that the magnitude function $\mathcal{M}_X$ “depends asymptotically continuously” on the domain $X\in CoCo_0^\infty(\mathbb{R}^n)$ in the Hausdorff metric up to order $N$?

Let us make this statement precise. For which $N$’s does it hold that for all $X\in CoCo_0^\infty(\mathbb{R}^n)$ and $\epsilon\gt 0$ there are $C,\delta\gt 0$ such that $\left| \left(\mathcal{M}_X(R)-\sum_{j=0}^N c_j(X) R^{n-j}\right)-\left(\mathcal{M}_A(R)-\sum_{j=0}^N c_j(X) R^{n-j}\right)\right|\leq C\epsilon R^{n-N},$ for $A\in CoCo_0^\infty(\mathbb{R}^n)$ within Hausdorff distance $\delta$ from $X$?

If Problem C has a positive solution for some $N$ and Problem B has a positive solution for $j=0,\ldots, N$, from a density argument and continuity we could deduce that for any $X\in CoCo(\mathbb{R}^n)$ there is an asymptotic expansion $\mathcal{M}_X(R) = \sum_{j=0}^N c_j(X) R^{n-j}+o(R^{n-N})$ as $R\to \infty$. Again by density, we would have an asymptotic inclusion/exclusion principle $\mathcal{M}_A+\mathcal{M}_B+\mathcal{M}_{A\cap B}=\mathcal{M}_{A\cup B}+o(R^{N-n}),$ for $A,B\in CoCo(\mathbb{R}^n)$ with $A\cup B$ convex. The results of Barceló-Carbery shows that $n!\omega_n\mathcal{M}_X(R)=\mathrm{vol}_n(X)+o(R^n)$ so Problem C has a positive solution at least for $N=0$. Since $c_3$ does not depend continuously on $X$, it is only interesting to consider $N\lt 3$.

Posted at January 23, 2019 11:49 AM UTC

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### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Thanks for this post! There are some very thought-provoking results here.

One small comment:

An important general result of Leinster–Meckes says that the magnitude $X \mapsto \mathrm{Mag}(X)$ depends continuously on compact convex domains $X$.

This is a special case of Theorem 4.15 of the survey paper by Mark Meckes and me. And it’s true, both our names are on the paper. But it’s certainly Mark’s result, not mine.

Posted by: Tom Leinster on January 23, 2019 1:07 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Actually, here’s how I remember it: I mentioned that it would be nice if we knew that fact for Euclidean space. And you said you thought it followed easily from my earlier theorem on the continuity of the magnitude function as a function of $R$. This turned out to be true, but I had never noticed it. So I’d say that’s definitely a joint theorem.

I suppose I deserve credit for the extension to more general quasinorms, but that’s not directly relevant to this post.

Posted by: Mark Meckes on January 23, 2019 1:40 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

I second Tom’s thanks. This is a very interesting post!

A comment on notation: in convex geometry, what you denote by $CoCo(\mathbb{R}^n)$ and $CoCo_0(\mathbb{R}^n)$ are more or less standardly denoted by $\mathcal{K}^n$ and $\mathcal{K}_0^n$. (The space $CoCo_0^\infty(\mathbb{R}^n)$ also comes up, but I don’t think convex geometers have a standard notation for it.)

Posted by: Mark Meckes on January 23, 2019 1:45 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Thanks Mark for this remark. This notation is completely my fault. I was unaware of any standard notation, so I just went for something reasonably self-explanatory (yet still ugly enough so that no one would think it standard).

Posted by: Magnus Goffeng on January 23, 2019 6:32 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Another brief comment: the fact that $\lim_{\varepsilon \to 1} \mathcal{M}_{X_\varepsilon}(R) = \mathcal{M}_{X_1}(R)$ follows from general regularity results, in this case that magnitude is increasing and lower semicontinuous as a function of $X$.

And a question: in the unpublished work you mention on a closed Riemannian surface with the geodesic distance, what do you take as the definition of magnitude?

Posted by: Mark Meckes on January 23, 2019 6:17 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

The definition we use for closed manifolds is that we look for a measure $\mu$ solving $Z_R \mu=1$ where $Z_R\mu(x)=\int_X \mathrm{e}^{-R\mathrm{d}(x,y)}\ \mathrm{d}\mu(y)$. In this special case, the solution $\mu$ will be of the form $f$ times Lebesgue measure for a smooth function $f$ by elliptic regularity. Moreover, $Z_R$ is invertible for large $\mathrm{Re}(R)$. I guess one can expect that the magnitude function has poles on the positive real axis in general, an example of that would have been nice to see.

Posted by: Magnus Goffeng on January 23, 2019 6:37 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Okay, so in our jargon for magnitude you’re using weight measures, and you prove that for sufficiently large real $R$, a weight measure exists. That’s a nice generalization (nicer than I hoped for!) of a basic fact we know about finite spaces.

Is your $f$ also nonnegative for sufficiently large real $R$?

Posted by: Mark Meckes on January 23, 2019 7:34 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

As far as I can remember, we have not thought about positivity of $Z_R^{-1}1$. Maybe one could prove something as $R\to \infty$, but I will have to think about it (maybe Heiko knows?). Why would it be interesting to know that $f$ is positive?

Posted by: Magnus Goffeng on January 23, 2019 8:13 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

For a finite space, we know the weighting is nonnegative for sufficiently large $R$. We also know that magnitude is continuous (with respect to the Gromov–Hausdorff topology) on the class of positive definite metric spaces with nonnegative weight measures. So I should also add the question of whether $Z_R$ defines a positive semidefinite quadratic form on finite signed measures for sufficiently large real $R$.

Posted by: Mark Meckes on January 23, 2019 8:43 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

As far as I see, in your language we prove that in the case of a closed Riemannian manifold the operator $Z_R$ defines a quadratic form on a Sobolev space of negative order, which is positive definite for large $R$. This is true in even and odd dimension, as long as the manifold is closed, i.e. there is no boundary.

Whether the function $f$ might be positive, requires some further thought.

Posted by: Heiko Gimperlein on January 23, 2019 10:33 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

Great. So in the terminology and notation Tom and I have used, if $(X,d)$ is a closed Riemannian manifold equipped with the geodesic distance $d$, then $t X$ is a positive definite metric space for sufficiently large $t$, and $t X$ possesses a weight measure for sufficiently large $t$. Can you quantify how large is sufficiently large in either of those statements?

(You may already know this, but for the benefit of interested bystanders, there do exist closed Riemannian manifolds $X$ for which $t X$ is fails to positive definite for some $t$; that is, $X$ is not of negative type. Examples and references are in section 3.2 of “Positive definite metric spaces”.)

For comparison, Proposition 2.4.17 of “The magnitude of metric spaces” implies that if $(X,d)$ is a metric space with $n$ points, then $t X$ is positive definite and possesses a nonnegative weighting for $t > \frac{\log (n-1)}{\min_{x\neq y} d(x,y)}.$

Posted by: Mark Meckes on January 24, 2019 2:48 PM | Permalink | Reply to this

### Re: On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective

An explicit criterion is not immediate from our approach, but we have not thought about it. Basically, using a Taylor expansion of the distance function $d(x,y)$ for $y\to x$, we prove that the operator $Z_R$ admits an expansion with leading term $(R^2-\Delta)^{-m}$. The bilinear form associated to this leading term is positive definite and will dominate the lower order terms for $R \to \infty$.

One can probably obtain an $R_0$ such that the bilinear form is positive definite for $R\geq R_0$ from an estimate for the error of the Taylor expansion of $d(x,y)$. This would seem to involve the curvature tensor, but the details would require some care.

Posted by: Heiko Gimperlein on January 24, 2019 3:37 PM | Permalink | Reply to this

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