# Convergence in the proof of Crofton's Formula

Let $\mathcal{L}$ be the set of oriented lines in $\mathbb{R}^2$ and let $\mu$ be the Kinematic Measure on $\mathcal{L}$; up to scaling, $\mu$ comes from the unique (up to scaling) volume form on $\mathcal{L}$ that is invariant under the action of the group of isometries of $\mathbb{R}^2$. Given any function $f:[0,1] \rightarrow \mathbb{R}^2$, define a function $n(f):\mathcal{L} \rightarrow \mathbb{Z} \cup \{\infty\}$ by setting $n(f)(L)$ to be the number of elements of the set $\{\text{$x \in [0,1]|f(x) \in L$}\}$.

The key step in one of the standard proofs of Crofton's formula asserts the following. Let $f:[0,1] \rightarrow \mathbb{R}^2$ be a rectifiable curve. For any partition $P$ of $[0,1]$, let $f_P:[0,1] \rightarrow \mathbb{R}^2$ be the associated polygonal approximation to $f$, so $f_P(x) = f(x)$ for $x \in P$ and $f_P$ restricts to a straight line segment on each component of $[0,1] \setminus P$. The set of partitions of $[0,1]$ is partially ordered by refinement, and by definition the length of $f$ is the limit (with respect to this partial order) of the lengths of the $f_P$.

The "key step" referred to above is the following: the limit (over the set partitions $P$ of $[0,1]$) of $\int_{\mathcal{L}} n(f_P) d\mu$ equals $\int_{\mathcal{L}} n(f) d\mu$.

I don't see how to prove this and cannot find a source that has a complete proof. Can anyone either give me a proof or a good reference for it?

• you can find a proof in the paper of S Ayari and S Dubuc : La formule de Cauchy sur la longeur d'une courbe ; Canadian Mathematical Bulletin vol 40 (1) 1997 pages 3-9 Oct 19, 2015 at 18:17

## 1 Answer

Rectifiability is a very weak condition, and the method of approximation of length you use is not the best one. Normally one uses circumscribed polygons rather than inscribed polygons as you suggest. For higher dimensional objects the famous example of Schwartz shows why the approach based on inscribed polyhedra fails spectacularly. In particular, the approach to Crofton formula you suggest does not work in higher dimensions.

The typical proof of Crofton formula relies on the double fibration trick aided by the coarea formula. Here is a proof of this formula for curves. The same proof extends to higher dimensions as well; see Chapter 9 of these notes.