### Integral Geometry in Barcelona

#### Posted by Tom Leinster

Simon Willerton and I are currently at the Advanced Course on Integral Geometry and Valuation Theory at the Centre de Recerca Matemàtica, Barcelona. Over the next couple of days we’re going to be presenting our own work, along with some work of Mark Meckes, on magnitude of metric spaces. You can read my slides here, and when I see Simon next I’ll hassle him to make his slides available too—if he hasn’t already got the hint.

I’m a beginner in integral geometry, which makes this meeting an adventure. Over the last couple of years I’ve been trying to educate myself in the subject a bit. In particular, I was in Barcelona for the week before the course, at the kind invitation of Joachim Kock. He’s also attending the course, so we spent some time trying to get the basics straight. I think it’s paying off.

Since I’m such a novice, I’m not going to attempt a running summary of the talks. But just to give the flavour of the meeting, I’ll say a little about each of the first day’s talks.

Semyon Alesker kicked off. He’s giving one of the two
courses that form the heart of the week’s activities. It’s an hour and a half
every day for five days, which was enough to
put the fear of God into me. I’d also had enough contact with the
subject to know that he’s a total hotshot, so I sat down in the lecture room
with some trepidation, suspecting I’d be rapidly lost. My fears weren’t calmed
when he began by explaining that he appreciated what a diverse audience we
were: he comes from *convex* geometry, whereas many of us come from
*integral* geometry. But it was a really nice lecture, paced slowly
enough that I could follow right the way through, and strikingly
well organized—in short, a pleasure.

The
CRM
also has the great procedure of getting its lecturers to produce notes
*beforehand*, so that when you sit down to hear Lecture 1, you already
have a printed
set of notes before you.

The other course is being given by Joseph Fu. I’d previously been reading a different set of lecture notes by him, on ‘Blaschkean integral geometry’, so it was good to see him in action. He talked a lot about how Alesker has revolutionized the subject (while being rather modest, I suspect, about his own contributions).

Among other things, Fu
spoke about what he calls the *template method*. This is, essentially,
what I described as ‘the grit around which the pearl forms’
back here,
when I was explaining a nice solution to the Buffon needle problem. The
general situation is as follows. In integral geometry, you often end up with
an equation of the form ‘$f(A) = g(A)$ for all $A$’ involving some constants.
You might not know what those constants
are, but you know that there are *some* constants for which the equation
holds. The template method is to stick in some particularly convenient value
or values of $A$ in order to discover those constants. But it’s a bit of an ad
hoc method, and can lead to tricky calculations… so I think he’s going to
suggest something better later in the week.

In the afternoon, Luis Cruz-Orive gave a talk about computations on subsets of $\mathbb{R}^3$, from an engineering perspective. We can argue endlessly about what constitutes concrete or abstract mathematics, or whether the word ‘abstract’ actually means anything. But this talk was about granular cementing: if that’s not concrete, what is?

Partway through the talk, Cruz-Orive stopped, turned dramatically to the
audience, and said something along the following lines. “You all prove general
theorems, and that’s good. But I urge you, I *urge* you, take a
break, take a sabbatical year, and spend a year applying your theorems in
dimension three.” So, $n$-categorists: spend a year applying your theorems to
tricategories. It’s not *such* inappropriate advice.

Eva Vedel Jensen gave the last talk, on rotational Crofton formulas. Now, Crofton’s formula is a really nice thing to explain, and I’d kind of like to explain it—it’s a generalization of that solution to Buffon’s needle problem that I explained before and mentioned above. It’s a staple of integral geometry. Unfortunately I don’t have time to tell the story now… but if anyone reading wants to have a go, please do!

Incidentally, Crofton published his work in an unusual place: the
*Encyclopaedia Britannica*. More specifically, it was the entry on Probability in the 9th
edition, 1885. In their book *Introduction to Geometric
Probability*, Klain and Rota describe it as

Crofton’s article in the ninth edition of the Encyclopaedia Britannica, an article that created the subject from scratch and that is still worth reading today.

I once looked for it online, thinking that it must be out of copyright and available, but couldn’t find it. If anyone else can find it, I’d love to know.

## Re: Integral Geometry in Barcelona

I’ve put my talk up:

The purpose of my talk was to give some justification to the claims made by Tom in his talk that the notion of magnitude (a.k.a. Euler characteristic or cardinality) of a metric spaces is somehow related to the ideas of intrinsic volumes. Let me remind you a flavour of what we believe to be one connection.

If $A$ is a finite subset of the Euclidean plane then we know that as a finite metric space this has a well defined magnitude $|A|$, given in terms of a weighting of the points. Suppose that $K$ is a compact, convex subset of the plane, for example a disc, a square or a triangle. Now suppose that $\{A_i\}_{i=0}^\infty$ is a sequence of finite subsets of $K$, so that $A_i\subset K$, and that the sequence ‘fills out’ the convex set $K$, in other words $A_i\to K$ in the Hausdorff topology (so every point in $K$ is eventually as close as you like to the subsets). The

Convex Conjecturesays that the magnitudes of the subsets converge to a combination of classical invariants of $K$:$If\quad A_i\to K\quad then\quad \left|A_i\right|\to \frac{1}{2\pi} Area(K) + \frac{1}{4} Perimeter(K) + 1.$

More generally, suppose that $K$ is a compact, convex subset of an arbitrary $n$-dimensional Euclidean space. Let $\mu_j(K)$ denote the $j$th intrinsic volume of $K$, so that $\mu_n(K)$ is the volume of $K$ and $\mu_{n-1}(K)$ is half the surface area of $K$. Let $\omega_j$ denote the volume of the unit $j$-ball. The

Convex Conjecturein this setting says:$If\quad A_i\to K\quad then\quad \left|A_i\right|\to \sum_{j=0}^n\frac{\mu_j(K)}{j!\,\omega_j} .$

As far as I know, we currently have no idea how to attempt to prove this. The justification for this conjecture is based on a small number of exact calculations, such as when $K$ is an interval or a point, and a number of large computations in small dimensions.

Before giving my talk I had a bit of a crisis of confidence. I thought I would be torn apart by the audience because of the small amount of evidence we have for this conjecture. I was concerned that I really should have tried more and more computations before standing up in front of such people and making such an assertion. Needless to say, they didn’t tear me apart for that.

After getting back from Barcelona I got the computer to do some more computations. Now I am even more convinced that the conjecture is true! (But even more frustrated by the lack of ideas on how to prove it…)