Besicovitch-Federer projection theorem for mappings having constant rank of the Jacobian matrix

TL;DR

This paper generalizes the Besicovitch-Federer projection theorem to mappings with constant Jacobian rank, showing how unrectifiable sets can be approximated by mappings with zero measure projections.

Contribution

It extends the projection theorem to smooth maps with constant Jacobian rank, providing approximation results for unrectifiable sets.

Findings

Existence of approximating mappings with zero measure projections

Generalization of the Besicovitch-Federer theorem to constant rank maps

Approximation in the ^1 topology for unrectifiable sets

Abstract

The purpose of this article is to prove a generalisation of the Besicovitch-Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an $m$-unrectifiable set $\Sigma\subset\mathbb{R}^n$ having finite Hausdorff measure and $\varepsilon>0$, we prove that for a mapping $f\in\mathcal{C}^1(U,\mathbb{R}^n)$ having constant, equal to $m$, rank of the Jacobian matrix there exists a mapping $f_\varepsilon$ whose rank of the Jacobian matrix is also constant, equal to $m$, such that $\|f_\varepsilon-f\|_{\mathcal{C}^1}<\varepsilon$ and $\mathcal{H}^m(f_\varepsilon(\Sigma))=0$. We derive it as a consequence of the Besicovitch-Federer theorem stating that the $\mathcal{H}^m$ measure of a generic projection of an $m$-unrectifiable set $\Sigma$ onto an $m$-dimensional plane is equal to zero.