This article is related to the following conjecture made by S. W. Semmes [Proceedings of the International Congress of Mathematicians, ICM ’94, Vol. II, 875-885 (1995; Zbl 0897.28003)].
Suppose that \(\Omega\) is a bounded set in \({\mathbf R}^n\), \(n>2\), and suppose that \(B(0,1)\subset \Omega\), \({\mathcal H}^{n-1}(\partial \Omega)=M<\infty\). Then there are \(\varepsilon >0\), \(L< \infty\) and a Lipschitz graph \(\Gamma\), with constant \(L\), such that \({\mathcal H}^{{n-1}(\Gamma \cap \partial \Omega)}\geq \varepsilon\).
This conjecture has been proved earlier by G. David and S. Semmes [Pac. J. Math. (to appear)]. Their result is obtained in a different way than in this paper. The method here is to use a result the authors call a Checkerboard Theorem, briefly described below. This is used to put together the result from a \(2\)-dimensional result that was previously known.
Let \(A,B\subset {\mathbf R}^{n}\), then \(A\) is said to be checkerboard connected through \(B\) if for any \(x,y\in A\) there is a path from \(x\) to \(y\) consisting of a finite number of line segments along one of the coordinate direction having both endpoints in \(B\). The checkerboard distance is the infimum over the length of such paths. The Checkerboard Theorem follows.
Given any \(\delta>0\) and a measurable set \(B\subset [0,1]^{n}\) with \(| B| =\varepsilon\), there exists a subset \(A\subset B\) with \(| A| \geq (1-\delta)\varepsilon^{n}\) and with \(A\) checkerboard connected through \(B\).
There are also other applications of this theorem; it is also used to prove parts of the “Structure Theorem” of geometric measure theory, and also to obtain a version of the Almgren’s tilt-excess theorem.