Menger curvature and rectifiability

J.C. Léger

Review from Zentralblatt MATH:

The background of the main theorem proved in the interesting paper under review lies in the well-known Vitushkin conjecture dealing with sets which are removable for bounded analytic functions. A compact set in the plane is called removable for bounded analytic functions if all bounded analytic functions on its complement are constant. Furthermore, a Borel set $E\subset\Bbb R^n$ is said to be rectifiable if there are Lipschitz functions $f_i:\Bbb R\to\Bbb R^n$, $i=1,2,\dots$, such that $\Cal H^1(E\setminus\bigcup_{i=1}^\infty f_i(\Bbb R))=0$. The set $E$ is purely unrectifiable if $\Cal H^1(E\cap f(\Bbb R))=0$ for any Lipschitz function $f:\Bbb R\to\Bbb R^n$. Here $\Cal H^1$ is the 1-dimensional Hausdorff measure.

According to the Vitushkin conjecture a compact subset of the plane with positive and finite 1-dimensional Hausdorff measure is removable for bounded analytic functions if and only if it is purely unrectifiable. The proof of the conjecture was completed by {\it G. David} in the remarkable paper [Rev. Mat. Iberoam. 14, No. 2, 369-479 (1998; Zbl 0930.30012)]. For detailed description of many important contributions done by several people in the process of solving the conjecture see the survey by {\it G. David} [Publ. Mat., Barc. 43, No. 1, 3-25 (1999)].

In the paper under review the author proves that a Borel set $E\subset\Bbb R^n$ with $0<\Cal H^1(E)<\infty$ is rectifiable provided that it has finite total Menger curvature, that is, $$c^2(E)=\int_E\int_E\int_Ec^2(x,y,z)\Cal H^1(x)\Cal H^1(y)\Cal H^1(z) <\infty,$$ where $c(x,y,z)$ is the inverse of the radius of the circle whose circumference passes through the points $x$, $y$, and $z$. This result is used by G. David in his proof of the Vitushkin conjecture and it was earlier proved by him in an unpublished paper. The author points out that, compared to the one given by G. David, his proof has the benefit that it is quite easy to extend to dimensions higher than 1.

Reviewed by Maarit Järvenpää

Keywords: Menger curvature; rectifiability; Vitushkin conjecture

Classification (MSC2000): 28A75 28A80

Full text of the article:

• PDF file (292 kilobytes)