The n-Category Café

In the beginning, there was the magnitude of finite metric spaces, a special case of the magnitude of finite enriched categories. Simon Willerton and I did some early investigation of how to extend magnitude from finite to infinite metric spaces — or more specifically, compact metric spaces. Interesting as the results of that investigation were — and we found out lots of cool stuff about the magnitude of spheres, Cantor sets, and so on — there was something ad hoc at its heart: the very definition of the magnitude of a compact space.

That foundational problem was largely settled by Mark in a pair of papers about ten years ago. They show definitively how to generalize magnitude from finite to compact metric spaces, at least under the mild technical condition that the spaces concerned are positive definite (which I won’t go into here). There are several different ways that one might imagine extending the definition of magnitude from finite to compact spaces, and Mark proved that under this hypothesis, they all give exactly the same result.

But some questions remained. A particularly prominent one was this. By definition, the magnitude of a nonempty positive definite compact metric space lies in the interval $[1, \infty]$. Given any $m \in [1, \infty)$, it’s easy to find a space with magnitude $m$. But is there anything with magnitude $\infty$?

This question has been open since 2010, but we settle it in our new paper. The answer is yes. One might have guessed that no such space exists: after all, compactness is a kind of finiteness condition, so perhaps it wouldn’t be surprising if it implied finiteness of magnitude. But our counterexample reveals the fuzziness in that thinking.

The counterexample is an infinite-dimensional simplex in sequence space $\ell^1$, as follows.

First imagine that we’ve chosen two positive real numbers, $a_1$ and $a_2$, and consider the convex hull of the points

$$(0, 0), \ (a_1, 0), \ (0, a_2)$$

in $\mathbb{R}^2$. It’s a 2-simplex. Similarly, given positive reals $a_1, a_2, a_3$, the convex hull of

$$(0, 0, 0), \ (a_1, 0, 0), \ (0, a_2, 0), (0, 0, a_3)$$

in $\mathbb{R}^3$ is a 3-simplex.

Now do the same thing in $\ell^1$ for an infinite sequence $a_1, a_2, \ldots$. Write $X$ for the convex hull, with the metric from the $\ell^1$ norm. (Or to be precise, it’s the closed convex hull.) Now:

• $X$ is compact $\iff$ $a_n \to 0$ as $n \to \infty$

• $X$ has finite magnitude $\iff$ $\sum_n a_n \lt \infty$.

So we get a compact space of infinite magnitude by taking any sequence $(a_n)$ of positive reals that converges to $0$ but has infinite sum.

In other words, the gap between compactness and finite magnitude is the same as the gap we emphasize when we teach a first course on sequences and series: for a series to converge is a stronger condition than for its sequence of entries to converge to $0$.

The second question we settle in our paper is about spaces of minimal magnitude. I mentioned earlier that the magnitude of a nonempty positive definite compact metric space (let me just say “space”) is in the interval $[1, \infty]$. We now know that there are spaces with magnitude exactly $\infty$. It’s also trivial — once you have the definitions! — that for any infinite space $X$, the magnitude $|t X|$ of the scaled-up space $t X$ converges to $\infty$ as $t \to \infty$. But what about the other end of the scale: spaces of magnitude close to or equal to $1$?

As it turns out, the situation for $1$ is the opposite way round from the one for $\infty$. What I mean is this. It’s easy to say which spaces have magnitude exactly $1$: it’s just the one-point space. Now given any space $X$, the rescaled space $t X$ looks more and more like a point as $t \to 0$, so you might expect that

$$\lim_{t \to 0} |t X| = 1.$$

But in fact, this is where the complexity lies. Long ago, Simon found an example of a 6-point space $X$ such that $\lim_{t \to 0} |t X| = 6/5$. So we have a nontrivial situation on our hands.

Say that a space $X$ has the one-point property if $\lim_{t \to 0} |t X| = 1$. Although not every space has the one-point property, there are lots that do. For instance, it’s not so hard to show that every compact subset of $\mathbb{R}^n$ with the taxicab metric (the metric induced by the 1-norm) has the one-point property. And with much more effort, it was shown that compact subsets of $\mathbb{R}^n$ with the Euclidean metric have the one-point property too, first by Juan Antonio Barceló and Tony Carbery, then, by different arguments, by Simon and by Mark.

In our new paper, we find a single sufficient condition that unifies these results:

Let $V$ be a finite-dimensional normed vector space that is positive definite as a metric space. Then every nonempty compact subset of $V$ has the one-point property.

For instance, we could take $V$ to be $\mathbb{R}^N$ with either the taxicab or Euclidean metric, or more generally, we could give it the metric induced by the $p$-norm for any $p \in [1, 2]$. An equivalent condition on $V$ is that it’s isometrically isomorphic to a linear subspace of $L^1[0, 1]$, which gives a 1-norm flavour to this theorem too.

So our paper consists of these two theorems: one on spaces with magnitude $\infty$, and one on spaces with magnitude close to $1$.

The proofs have something in common. Longtime Café readers will remember that around the time when magnitude got going, Simon and I made a conjecture about the magnitude of convex subsets of Euclidean space. That conjecture turned out to be false, although several aspects of it were correct.

But less publicized was an analogous conjecture for convex subsets of $\mathbb{R}^N$ with the taxicab metric. And this conjecture turned out to be true! Or at least, true when the convex set $X$ has nonempty interior. The result is that

$$|X| = \sum_{i = 0}^N 2^{-i} V'_i(X)$$

(Theorem 4.6 here), where $V'_i$ is a 1-norm analogue of the $i$th intrinsic volume. For example, this implies that the magnitude function $|t X|$ of $X$ is a polynomial:

$$|t X| = \sum_{i = 0}^N 2^{-i} V'_i(X) \cdot t^i.$$

And even if $X$ has empty interior, we still have an inequality one way round: $|X| \leq \sum \ldots$. (It may be an equality for all we know — that remains unsettled.)

In any case, this result gets used in the proofs of both theorems in our new paper: first, to find the magnitude of that infinite-dimensional simplex, and second, to bound the magnitude of polytopes, which turns out to be the key to the whole thing.

By now, I feel like this post must already be nearly as long as the paper itself, so I’ll link to it once more and stop.