## September 6, 2010

### 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.

Posted at September 6, 2010 11:14 PM UTC

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### 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 Conjecture says 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 Conjecture in 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…)

Posted by: Simon Willerton on September 16, 2010 11:06 AM | Permalink | Reply to this

### Re: Integral Geometry in Barcelona

There’s one more piece of evidence for the Convex Conjecture than Simon mentions here. It’s that there’s an analogous conjecture for $\ell_1^n$ (that is, $\mathbb{R}^n$ with the $\ell_1$ or $\ell^1$ metric), and we know that in a large number of cases.

The $\ell^1$ version is exactly the same, except that the terms involved all have to be interpreted in a suitable $\ell^1$ sense:

1. $\omega_j$, the volume of the unit $j$-ball, changes because the ball has changed; its value is in fact $2^j/j!$
2. For reasons I won’t explain right now, the $j$th intrinsic volume $\mu_j$ is defined as follows: $\mu_j(K) = \sum_P Vol(\pi_P(K))$ where $P$ ranges over all $j$-element subsets of $\{1, \ldots, n\}$, where $\pi_P: \mathbb{R}^n \to \mathbb{R}^j$ is projection onto the corresponding $j$ coordinates, and where $Vol$ is Lebesgue measure.
3. ‘Convex’ can be interpreted in the usual sense, but I believe it can be weakened to the following condition: a subset $K \subseteq \mathbb{R}^n$ is $\ell^1$-convex if for all $a, b \in K$, there exists an isometry $\gamma: [0, d(a, b)] \to K$ such that $\gamma(0) = a$ and $\gamma(d(a, b)) = b$.

I think I know how to prove this $\ell^1$ convex conjecture, at least in the case where $K$ is the closure of its interior. By ‘know how to prove’ I mean I have a fairly detailed plan; but I haven’t executed it yet.

I do claim that I can prove it in many more cases than we know for the $\ell^2$ (Euclidean) case. For example, it’s very easy when $K$ is a cuboid $\prod_i [a_i, b_i]$ with edges parallel to the coordinate axes.

However, the techniques I’m using for $\ell_1^n$ use something very special about $\ell^1$: basically, the fact that the $\ell^1$ product is the tensor product of enriched categories. I have no idea how they could be transferred to $\ell_p^n$ for any other value of $p$ (in particular, $2$).

Posted by: Tom Leinster on September 16, 2010 7:15 PM | Permalink | Reply to this

### Re: Integral Geometry in Barcelona

If you have no idea what integral geometry is and want a taste, have a look at this page on John’s blog—particularly my comments.

Posted by: Tom Leinster on October 16, 2010 9:51 PM | Permalink | Reply to this

### Re: Integral Geometry in Barcelona

Last week I gave a talk on magnitude at a workshop on Geometric Probability and Optimal Transportation at the Fields Institute. Here are slides and audio. I only had 25 minutes, as opposed to Tom and Simon who I think had 50 minutes each in Barcelona, but nevertheless squeezed in a few details that weren’t in either of their talks.

Posted by: Mark Meckes on November 13, 2010 1:37 AM | Permalink | Reply to this

### Re: Integral Geometry in Barcelona

Thanks, Mark. There is indeed a lot there that neither Simon nor I said in Barcelona. You’re showing your skill at optimization.

Is the audio available? I can’t find it.

Posted by: Tom Leinster on November 13, 2010 3:12 PM | Permalink | Reply to this

### Re: Integral Geometry in Barcelona

There is supposed to be audio, I think; I had to wear a microphone for it. They only posted the slides yesterday, so maybe the audio will appear later, or maybe there was some problem with it.

Posted by: Mark Meckes on November 13, 2010 6:06 PM | Permalink | Reply to this

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