## Integral Geometry

Classical integral geometry applies to smooth, or at least rectifiable sets that are anything but of a fractal nature. A fundamental formula is due to Poincaré: if $E$ and $F$ are rectifiable curves in the Euclidean plane then
\begin{aligned} \int(\text { number of points in } E \cap \sigma(F)) d \sigma & \ & =4(\text { length of } E)(\text { length of } F) \end{aligned}
where integration is with respect to the natural invariant measure on the set of rigid motions $\sigma$ (thus $d \sigma=d \theta d x d y$ where $(x, y)$ is the point of the plane containing $E$ to which an end of $F$ is translated, and $\theta$ is the angle of rotation). Poincaré’s formula is easily verified, first for a pair of line segments, then by polygonal approximation to general $E$ and $F$. A similar idea applies if smooth manifolds $E$ and $F$ in $\mathbf{R}^n$ are moved rigidly relatively to each other. If they intersect at all, then ‘in general’ they will intersect in a set of $\operatorname{dimension} \max {0, \operatorname{dim} E+\operatorname{dim} F-n}$. More precisely, if $\operatorname{dim} E+$ $\operatorname{dim} F-n>0$ then $\operatorname{dim}(E \cap \sigma(F))=\operatorname{dim} E+\operatorname{dim} F-n$ for a set of rigid motions $\sigma$ of positive measure, and is 0 for almost all other $\sigma$.

The problem we consider here is whether such results hold if $E$ and $F$ are fractals and we use Hausdorff dimension: for which $E, F \subset \mathbf{R}^n$ do we have
$$\operatorname{dim}(E \cap \sigma(F)) \leq \max {0, \operatorname{dim} E+\operatorname{dim} F-n} \text { ‘in general’ (A) }$$
and

$$\operatorname{dim}(E \cap \sigma(F)) \geq \operatorname{dim} E+\operatorname{dim} F-n \text { ‘often’ (B) }$$
as $\sigma$ ranges over an appropriate group $G$ of transformations (e.g. translations, congruences or similitudes)? Of course, ‘in general’ means for almost all $\sigma$, and ‘often’ means for a set of $\sigma$ of positive measure, with respect to the natural invariant measure on $G$. We restrict our discussion of the main results to the case when $E$ and $F$ are ‘reasonable’ sets such as Borel or compact sets such as occur in practice – otherwise very few positive results are possible! Detailed proofs are not given here, except in a couple of cases to try to indicate the ‘flavour’ of the subject.

## Towards Inequality A and B

As far as inequality $A$ is concerned, all the positive results that there are hold when $G$ is the group of translations, and so hold automatically for the larger groups of congruences and similarities. The following special case leads to the most general result. If $x \in \mathbf{R}^n$ and $F \subset \mathbf{R}^n$, we write $x+F$ for the vector sum ${x+f$ : $f \in F}$.

Lemma 1. Let $E, F \subset \mathbf{R}^2$ with $F$ a line segment. Let $1 \leq \operatorname{dim} E \leq 2$. Then
$$\operatorname{dim}(E \cap(x+F)) \leq \operatorname{dim} E-1=\operatorname{dim} E+\operatorname{dim} F-2$$
for almost all $x \in \mathbf{R}^2$.

Notice that a very similar argument works if $F$ is any rectifiable curve. In the same way, if $E, F \subset \mathbf{R}^n$ and $F$ is a $d$-dimensional flat (i.e. a subset of positive $d$-dimensional area of a translate of a $d$-dimensional subspace), then
$$\operatorname{dim}(E \cap(x+F)) \geq \max {0, \operatorname{dim} E+\operatorname{dim} F-n}$$
for almost all $x \in \mathbf{R}^n$.
We now use Lemma 1 to obtain a more general result.

Theorem 1. Theorem 1. Let $E, F \subset \mathbf{R}^n$. Then
$$\operatorname{dim}(E \cap(x+F)) \geq \max {0, \operatorname{dim}(E \times F)-n}$$
for almost all $x \in \mathbf{R}^n$.

Proof. We show this in the case of $E, F \subset R$; the result follows for $n>1$ in exactly the same way. Let $L$ be the line $y=x$ in $\mathbf{R}^2$. Assuming that $\operatorname{dim}(E \times F)>1$ we have that
$$\operatorname{dim}((E \times F) \cap((x,-x)+L))<\operatorname{dim}(E \times F)-1$$
for almost all $x \in \mathbf{R}$ by Lemma 1 . (Here we make use of the redundancy of one of the coordinate dimensions as we translate $L$ along itself.) But $(e, f) \in(E \times F) \cap((x,-x)+L)$ if and only if $e \in E, f \in F, e=x+y$ and $f=-x+y$ for some $y \in \mathbf{R}$, that is if $e=2 x+f$ or $e \in E \cap(2 x+F)$. Projecting onto the $x$-axis we see that $(E \times F) \cap((x,-x)+L)$ and $E \cap(2 x+F)$ are geometrically similar, so that (2) follows from (3) if $n=1$.

Although is a very long way from (A), examples show that it is the best result we can hope to achieve. In general for any $E$, F
$$\operatorname{dim}(E \times F) \geq \operatorname{dim} E+\operatorname{dim} F$$
However, in many situations we do have equality in (4), in which case (2) reduces to (A). This happens, for example, if either $E$ or $F$ is a rectifiable curve, or under more general conditions given terms of packing dimensions.

Lower estimates for $\operatorname{dim}(E \cap \sigma(F))$ are generally harder to obtain. The following is known:

Theorem 2. Let $E, F \subset \mathbf{R}^n$ be Borel sets, and $G$ a group of transformations on $\mathbf{R}^n$. Then
$$\operatorname{dim}(E \cap \sigma(F)) \geq \operatorname{dim} E+\operatorname{dim} F-n$$
for a set of motions $\sigma \in G$ of positive measure in the following cases:
(i) $G$ is the group of similarities and $E$ and $F$ are arbitrary.
(ii) $G$ is the group of congruences, $E$ is arbitrary and $F$ is a rectifiable curve or surface.
(iii) $G$ is the group of congruences and $E$ and $F$ are arbitrary with either $\operatorname{dim} E>\frac{1}{2}(n+1)$ or $\operatorname{dim} F>\frac{1}{2}(n+1)$.

We omit the proofs, which are based on the potential theoretic characterisation of Hausdorff dimension: if $E$ supports a mass distribution $\mu$ with $0<\mu(E)<\infty$ such that
$$\iint_E|x-y|^{-3} d \mu(x) d \mu(y)<\infty$$
then $\operatorname{dim} E \geq s$, and, conversely, if $s<\operatorname{dim} E$ then there exists such a mass distribution on $E$ with
$$\iint_E|x-y|^{-\bullet} d \mu(x) d \mu(y)<\infty$$

## Applications to Brownian Motion

Let $\omega(t)$ be a Brownian motion path in $\mathbf{R}^2$, so that the increments of the paths $\omega\left(t_2\right)-\omega\left(t_1\right), \ldots, \omega\left(t_{2 m}\right)-\omega\left(t_{2 m-1}\right)$ are independent if $t_1<t_2<\ldots<t_{2 m-1}<t_{2 m}$, and $\omega(t+h)-\omega(t)$ has zero (vector) mean and variance $h$ for all $t$. It has been known for a long time (Taylor [18]) that almost surely the path $\omega(t)$ has Hausdorff dimension 2. By Theorem 2 there exists a set of congruences $\sigma$ of positive measure such that $\operatorname{dim}(\omega([0,1]) \cap \sigma(\omega([2,3])))=2$. By the isotropy of Brownian motion, given the point $\omega(2)$ and a realization of $\omega([2,3])$, this realization is equally likely to occur rotated about $\omega(2)$ in any direction. Moreover, given $\omega([0,1])$, the distribution of the position of $\omega(2)$ is absolutely continuous with respect to Lebesgue measure. It follows that there is a positive probability of $\omega([0,1]) \cap \omega([2,3])$ having dimension 2 . By the selfsimilarity of the Brownian process, this same probability pertains in any time interval, so we conclude that the set of points on the path visited more than once has dimension 2 almost surely.

By repeating this argument, it follows that the set of points of multiplicity at least $m$ has dimension 2 almost surely for every positive integer $m$. Similarly, Theorems 1 and 2 may be used to show that almost surely Brownian paths in $\mathbf{R}^3$ have double points but no triple points, and that Brownian motion paths in $\mathbf{R}^n(n>4)$ have no multiple points.

## 下面是一些经典的积分几何The integral geometry题目

Problem 1.

Given $\omega \in \mathbf{S}^{n-1}, p \in \mathbf{R}$ let $T_{p, \omega}$ denote the distribution on $\mathbf{R}^n$ given by
$$T_{p, \omega}(f)=\widehat{f}(\omega, p) .$$
Then $T_{p, \omega}$ has support in the hyperplane $\langle\omega, x\rangle=p$ and $\frac{d}{d p}\left(T_{p, \omega}\right)$ is the normal derivative of this distribution. For $p=0$ we write this as $\frac{\partial}{\partial \nu} T_\omega$. For $n$ odd Theorem 3.8 can be written
$$\delta=c \int_{\mathbf{S}^{n-1}}\left(\frac{\partial^{n-1}}{\partial \nu^{n-1}} T_\omega\right) d \omega$$
which is a decomposition of the delta function into plane supported distributions.

Problem 2.

When parametrizing the set of lines in $\mathbf{R}^2$ by using $u x+v y=1$, the $\mathbf{M}(2)$ invariant measure is given by
$$\frac{d u d v}{\left(u^2+v^2\right)^{3 / 2}} \text {. }$$

Problem 3.

Let $\mathbf{F}_q$ be a finite field and $\mathbf{F}_q^n$ the $n$-dimensional vector space with its natural basis. The Hamming metric is the distance $d$ given by $d(x, y)=$ number of distinct coordinate positions in $x$ and $y$.

A linear $[n, k, d]$-code $C$ is a $k$-dimensional subspace of $\mathbf{F}k^n$ such that $d(x, y) \geq d$ for all $x, y \in C$. Let $\mathbf{P} C$ be the projectivization of $C$ on which the projective group $G=\mathbf{P G L}\left(k-1, \mathbf{F}_q\right)$ acts transitively. Let $\ell \in \mathbf{P} C$ be fixed and $\pi$ a hyperplane containing $\ell$. Let $K$ and $H$ be the corresponding isotropy groups. Then $X=G / K$, and $\Xi=G / H$ satisfy Lemma 1.3 and the transforms $$\widehat{f}(\xi)=\sum{x \in \xi} f(x), \quad \check{\varphi}(x)=\sum_{\xi \ni x} \varphi(\xi)$$
are well defined. They are inverted as follows. Put
$$s(\varphi)=\sum_{\xi \in \Xi} \varphi(\xi), \quad \sigma(f)=\sum_{x \in X} f(x) .$$The projective space $\mathbf{P}^m$ over $\mathbf{F}_q$ has a number of points equal to $p_m=$ $\frac{q^{m+1}-1}{q-1}$. Here $m=k-1$ and we consider the operators $D$ and $\Delta$ given by
$$(D \varphi)(\xi)=\varphi(\xi)-\frac{q^{k-2}-1}{\left(q^{k-1}-1\right)^2} s(\varphi), \quad(\Delta f)(x)=f(x)-\frac{q^{k-2}-1}{\left(q^{k-1}-1\right)^2} \sigma(f) .$$
Then
$$f(x)=\frac{1}{q^{k-2}}(D \widehat{f})^{\vee}(x), \quad \varphi(\xi)=\frac{1}{q^{k-2}}(\Delta \check{\varphi}) \curlyvee(\xi) .$$

Problem 4. According to Richter the equation $\lambda^{\prime}(D) \psi=0$ in Gonzalez’ Theorem characterizes the range of the X-ray transform on $\mathbf{R}^3$. Relate this to John’s equation $\Lambda \psi=0$

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