HAUSDORFF MEASURES OF SETS WHICH INTERSECT SUFFICIENTLY MANY LINEAR SUBSPACES

Abstract

In this paper we obtain an estimate for the $k$-dimensional Hausdorff measure of a set on the boundary of a domain $D$ in ${ℝ}^n$ intersecting each $(n-k)$-dimensional linear subspace in at least 2 points. To do this we prove a version of Crofton formula, which allows us to reduce the problem to the question about the existence of tangent planes and derivatives. In general, a Borel set may not admit tangent planes at any of its points. We introduce the notion of density tangent planes and prove that every set with the finite $k$-dimensional Hausdorff measure has density tangent planes at almost all of its points. We compute derivatives of some set functions at such points and then use the Crofton formula to get the final result.

It follows from the obtained estimate that the $k$-dimensional Hausdorff measure of such a set is not less than the minimal area of sections of $\partial D$ by $k+1$-dimensional linear subspaces when $D$ is a ball or a cube. This purely real problem has a complex application. The $1$-dimensional Hausdorff measure of a set on the boundary of the unit ball or the unit cube in ${ℂ}^n$ with its polynomial hull containing the origin is not less than $2\pi$ or $4$, respectively.