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Abstract

Let $m, n$ be positive integers such that $m< n$ and let $G(n,m)$ be the Grassmann manifold of all $m$-dimensional subspaces of $\mathbb{R}^n$. For $V\in G(n,m)$ let $\pi_V$ denote the orthogonal projection from $\mathbb{R}^n$ onto $V$. The following characterization of purely unrectifiable sets holds. Let $A$ be an $\mathcal H^m$-measurable subset of $\mathbb{R}^n$ with $\mathcal H^m(A)<\infty$. Then $A$ is purely $m$-unrectifiable if and only if there exists a null subset $Z$ of the universal bundle $\{ (V,v) \mid V\in G(n,m),\, v\in V\}$ such that, for all $P\in A$, one has $\mathcal H^{m(n-m)}(\{ V\in G(n,m) \mid (V,\pi_V(P))\in Z\})>0$. One can replace “for all $P\in A$” by “for $\mathcal H^m$-a.e. $P\in A$”.