## November 15, 2018

### Magnitude: A Bibliography

#### Posted by Tom Leinster

I’ve just done something I’ve been meaning to do for ages: compiled a bibliography of all the publications on magnitude that I know about. More people have written about it than I’d realized!

This isn’t an exercise in citation-gathering; I’ve only included a paper if magnitude is the central subject or a major theme.

I’ve included works on magnitude of ordinary, un-enriched, categories, in which context magnitude is usually called Euler characteristic. But I haven’t included works on the diversity measures that are closely related to magnitude.

Enjoy! And let me know in the comments if I’ve missed anything.

Posted at November 15, 2018 12:31 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3074

### Re: Magnitude: A Bibliography

One notable feature: of the 38 papers listed, 100% of them are on the arXiv.

Posted by: Tom Leinster on November 15, 2018 12:55 AM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

Splendid thanks! I was just trying to cobble something together yesterday for my PhD student from various Café posts.

Posted by: Simon Willerton on November 15, 2018 8:17 AM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

Nice idea! I hadn’t been aware of all of these.

Have you considered annotating them in any way? It’s not always the case that a paper’s most important contribution (in retrospect) is apparent from its abstract, so a sentence here or there about what an article establishes might be helpful. This thread could be a good place to gather suggestions for that, too.

(Unfortunately for me, I’ve only ever published/posted things about the diversity measures, rather than magnitude proper, so I can’t use this as an opportunity for self-promotion.)

Posted by: Blake Stacey on November 16, 2018 8:04 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

I just added a few more recent papers on magnitude to the bibliography. In an ideal world I’d write an explanatory blog post about each, but in the actual world, I’ll just list them here:

• Kiyonori Gomi, Magnitude homology of geodesic space. arXiv:1902.07044, 2019.

• Yasuhiko Asao, Magnitude homology of geodesic metric spaces with an upper curvature bound. arXiv:1903.11794, 2019.

• Mark Meckes, On the magnitude and intrinsic volumes of a convex body in Euclidean space. arXiv:1904.08923, 2019.

The one I understand best is the last, by Mark Meckes. The way I’d explain it is with reference to a false conjecture that Simon Willerton and I made long ago, which has nevertheless turned out to have many true aspects.

Simon and I guessed that for convex compact subsets $K$ of Euclidean space $\mathbb{R}^n$, the magnitude function $t \mapsto |t K|$ is a polynomial:

$|t K| = \sum_{i = 0}^n \frac{1}{i!\omega_i} V_i(K) t^i,$

where $V_i$ is the $i$-dimensional intrinsic volume and $\omega_i$ is the volume of the unit $i$-dimensional ball. For example, if $n = 2$, the conjecture says that the magnitude function $t \mapsto |t K|$ of a planar convex set $K$ is a quadratic whose constant term is $1$ (or $0$ if the set is empty), whose coefficient of $t^1$ is $1/4$ times the perimeter, and whose coefficient of $t^2$ is $1/2\pi$ times the area.

This turned out to be false. The first counterexample, provided by Juan Antonio Barceló and Tony Carbery, was the 5-dimensional Euclidean ball, whose magnitude function turns out to be a rational function of $t$ that isn’t a polynomial. But as I said a couple of paragraphs ago, there are some respects in which the conjecture makes correct predictions.

Mark has provided a couple more. The first is that, as he proves,

$|t K| \leq \sum_{i = 0}^n \frac{\omega_i}{4^i} V_i(K) t^i$

for convex sets $K$ in $\mathbb{R}^n$. We already knew that the magnitude function wasn’t a polynomial. And in fact, the example of the 5-ball showed that the conjectured polynomial was too small. But Mark has provided a formally similar, explicit, polynomial that is an upper bound.

That’s the first of the main two results in Mark’s paper. The second is to do with the “$t^1$” term in the magnitude function of balls. It doesn’t really make sense to speak of the $t^1$ term, since the magnitude function is not a polynomial. But Mark proves that for odd-dimensional Euclidean balls $B$,

$\lim_{t \to 0} \frac{|t B| - 1}{t} = V_1(B)/2.$

Up to a known constant, the 1-dimensional intrinsic volume of a Euclidean ball is just its radius. If $|t B|$ was a polynomial in $t$, this result would say that its coefficient of $t^1$ was the radius of $B$ (up to a known constant). That’s because we know that $\lim_{t \to 0} |t B|$ (which would be the constant term if it was a polynomial) is $1$.

This result was conjectured by Simon Willerton, and Mark makes the further natural conjecture that the same result holds for all compact convex sets, not just odd-dimensional balls.

Results like this are complementary to recent theorems by Heiko Gimperlein and Magnus Goffeng, blogged about recently here. They’ve found out lots about the top terms of the asymptotic expansion of $|t K|$, showing that, for instance, the first two terms give the volume and surface area (or its higher-dimensional analogue) of $K$. Mark’s second result here concerns the other end of the expansion.

Posted by: Tom Leinster on April 22, 2019 11:57 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

Thanks for the publicity! I’ll use this as as good a place as any to make a couple comments about my paper.

The first is a mathematical point that I realize in hindsight I should have included in the paper. (I’ll try to remember to add it whenever I next revise it.) By Heiko and Magnus’s work which you mentioned, if $K$ is a smooth compact domain in $\mathbb{R}^n$ and $n$ is odd, then the magnitude function $t \mapsto |t K|$ has a meromorphic continuation to the complex plane. Since we know $| t K | \to 1$ as $t \to 0^+$, the magnitude function is analytic, and hence has a power series expansion, in a neighborhood of $0$. So at least for those sets, it really does make sense to speak of the $t^1$ term!

Posted by: Mark Meckes on April 23, 2019 4:32 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

the magnitude function is analytic, and hence has a power series expansion, in a neighborhood of $0$. So at least for those sets, it really does make sense to speak of the $t^1$ term!

That’s excellent!

Do you have any thoughts about the $t^2$ term?

Posted by: Tom Leinster on April 25, 2019 8:03 AM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

I think it should be possible to compute the $t^2$ term for odd-dimensional balls by the same methods. Certainly the $t^2$ term could be expressed by a combinatorial sum using Simon’s formula, and I’m guessing the resulting sum could be evaluated by similar basic tricks.

But I think the complexity of approaching the $t^k$ term in this way probably increases exponentially with $k$, and that the $t^3$ term would already push the limits of what a normal human would have the patience to do by hand. Analyzing the $t^k$ term for arbitrary $k$ would require a new idea.

As for more general bodies, I don’t see a way forward on the $t^2$ term without nailing down the $t^1$ term more precisely first.

Posted by: Mark Meckes on April 25, 2019 1:27 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

Of course the obvious starting point for the $t^2$ term is to see what computer computations for low-ish odd $n$ say for the ball…

Posted by: Mark Meckes on April 25, 2019 1:32 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

The second comment is a more personal note: this was probably the most fun I’ve had writing a paper!

When I got started on magnitude, it was a bit of a departure for me. That was actually partly what attracted me to the topic: On the one hand, I was entertained to find a problem that I, as an analyst, was equipped to contribute to that was somehow connected with category theory. On the other hand it gave me an excuse to improve my knowledge of some important analytic tools (especially Fourier analysis) that hadn’t played a big role in my previous work.

That previous work was focused on the intersection of probability on the one hand, and convex geometry and the theory of finite-dimensional Banach spaces on the other hand. For example, the first paper I wrote was about volumes of convex hulls of random points in symmetric convex bodies. It involved some explicit calculations for unit balls of $\ell^p$ norms; the technical details involved combinatorial gadgets like generating functions, and probabilistic gadgets like the exact distribution of the determinant of a Gaussian random matrix. Although it’s all “analysis” in some sufficiently broad sense, it’s pretty different in character from the measure theory and Fourier analysis that are the main tools I’ve been using to study magnitude.

But, as it turns out, you always come back home. The first main result of this paper follows from an analogous result for $\ell^1$ spaces proved by Tom (which appears in our survey paper). To get from $\ell^1$ to $\ell^2$, I used the fact that a finite-dimensional $\ell^2$ can be almost isometrically embedded in an $\ell^1$ space, and that one can do this explicitly using probabilistic tools. The computations that translate the $\ell^1$ result into an $\ell^2$ result turn out to need … the distribution of the determinant of a Gaussian random matrix! The second main result (the one Simon conjectured) was proved using an exact combinatorial formula for $| t B |$ that Simon proved. Simon’s formula yields an expression for that $t^1$ term as a slightly ugly combinatorial sum. To simplify it to the point that it can be recognized as $V_1(B) / 2$, I had to brush off my long-neglected knowledge of … generating functions!

All in all, this had the feeling of going back to visit old friends, and telling them about all the things you’ve been up to since the last time you saw them.

Posted by: Mark Meckes on April 23, 2019 5:04 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

This is a really nice story.

When I was involved in running the graduate training programme for Scottish PhD students, I used to give an annual talk about the unexpected benefits of knowing a wide and unusual collection of things. I included lots of historical episodes, personal anecdotes, etc. This would have made a good addition!

Posted by: Tom Leinster on April 25, 2019 2:54 PM | Permalink | Reply to this

### Re: Magnitude: A Bibliography

I’ve just posted an update to this paper of mine. The main change is a new corollary of the upper bound on magnitude, which gives the first known examples of infinite-dimensional compact subsets of a Hilbert space with finite magnitude. (It’s still an open question whether every compact positive definite metric space has finite magnitude.)

The basic idea is that intrinsic volumes can be extended to infinite dimensions. If $K$ is a compact convex set in a Hilbert space, then $V_i(K) := \sup \{ V_i(L) \mid L \subseteq K \text{ is a finite-dimensional compact convex set}\}.$ It turns out that if $V_1(K) \lt \infty$ then $V_i(K) \lt \infty$ for every $i$. Compact convex sets $K$ with $V_1(K) \lt \infty$ are called GB bodies, which stands for “Gaussian bounded”, due to a relationship with boundedness properties of Gaussian random processes (another old friend of mine!).

Anyway, the upper bound that Tom stated implies that if $K$ is a GB body, then $|K| \lt \infty$. It also implies that $|t K | \to 1$ as $t \to 0$. The latter is a property that we already knew for finite-dimensional convex bodies, but it is not satisfied by all nice compact metric spaces.

Posted by: Mark Meckes on May 16, 2019 1:19 PM | Permalink | Reply to this

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