The n-Category Café

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 $N\lt 3$.