### What is Integral Geometry?

#### Posted by Tom Leinster

Spurred by an interest in metric spaces, I also got interested in integral geometry. But until quite recently I had no idea what it was. I’m guessing some of you have no idea either. The aim of this post is to give you a fast, very concrete explanation.

Here are three problems.

- Let $X$ be a cube in $\mathbb{R}^3$. What is the expected area of the orthogonal projection of $X$ onto a random plane through the origin?
- Let $X$ be a cube in $\mathbb{R}^3$, and suppose that $X$ sits inside some bigger cube $\hat{X}$. What is the probability that a random straight line in $\mathbb{R}^3$ (not necessarily through the origin) meets $X$, given that it meets $\hat{X}$?
- Let $X$ be a cube in $\mathbb{R}^3$, again sitting inside a bigger cube $\hat{X}$. Place a ball $Y$ of unit radius somewhere at random in $\mathbb{R}^3$, choosing uniformly among all positions such that $Y$ meets $\hat{X}$. What is the probability that $Y$ meets $X$?

You could probably figure out answers by a direct attack involving lots of integration. Things would get pretty ugly.

But integral geometry provides shockingly easy answers to all three questions—without doing any integrals!

Of course, it’s not just about these specific questions. For a start, the cubes $X$ and $\hat{X}$ and the ball $Y$ could be replaced by any compact convex sets, and the answers would still be easy. The space $\mathbb{R}^3$ could be replaced by any $\mathbb{R}^n$, and the lines and planes could be replaced by subspaces of any dimension. And there are other questions you can ask, though these are quite typical.

The formulas supplying the answers to the questions are called, respectively, Cauchy’s formula (or the mean projection formula), Crofton’s formula, and the kinematic formula. In fact, questions 1 and 2 are so closely related that there’s some blurring of the names; can you see why?

What makes these questions so shockingly easy to answer—once you know how—is the existence of some invariants called *intrinsic volumes*. I’ll explain it just for convex subsets of $\mathbb{R}^3$. Exact answers to all three questions can be given if you just have the following information about each of the spaces $X$, $\hat{X}$ and $Y$:

- the volume
- the surface area
- the mean width
- whether it’s empty.

(The *mean width* of a convex subset of $\mathbb{R}^3$ is its expected width, say in the direction of the $x$-axis, when it is oriented randomly. For example, the mean width of a ball is its diameter.) The formulas I named give answers to the questions in terms of these four invariants alone.

For example, the answer to the first question is ‘one quarter of the surface area of $X$’. We used this over at Azimuth to help with a problem about solar radiation.

## Re: What is Integral Geometry?

One of my favorite applications of integral geometry-type invariants is MacPherson and Srolovitz’s generalization of von Neumann’s law for grain growth to higher dimensions.

Roughly, one should think of a foam or the pattern of grains in a metal or other crystalline material (there’s a difference between foams and grains, but this is unimportant for the current purposes). The boundaries between the grains or the films in the foam evolve in time with normal speed proportional to the local curvature, see e.g. this experimental movie or this simulation.

von Neumann showed for 2D grains using a simple Gauss-Bonnet type argument that the rate of area change of each cell depends only on the number of sides minus 6; i.e. the rate of grain growth is topological!

Much effort went into trying to find an appropriate generalization, until MacPherson and Srolovitz showed that the proper generalization in 3D was that the rate of volume change was proportional to a quantity related to the mean width, and in higher dimensions, other integral geometric invariants! Check out the paper and supplementary information if you’re interested. It’s very readable.