Michigan Math. J. Advance Publication, 1-24, (2021) DOI: 10.1307/mmj/20205874
KEYWORDS: 28A12, 28A75, 28A05, 26Bxx, 26A45

Consider an integer $\mathit{n}\ge 2$, $\mathit{m}\in [\mathit{n},+\infty )$ and put $\stackrel{\u203e}{\mathit{\alpha}}:=\frac{\mathit{m}-1}{\mathit{n}-1}$. Moreover:

∙ Let ${\mathit{\gamma}}_{\mathit{x}}:(0,+\infty )\to {\mathbb{R}}^{\mathit{n}}$ be the line through $\mathit{x}=({\mathit{x}}_{1},\dots ,{\mathit{x}}_{\mathit{n}})\in {\mathbb{R}}^{\mathit{n}}$ defined as ${\mathit{\gamma}}_{\mathit{x}}(\mathit{t}):=(\mathit{t}{\mathit{x}}_{1},{\mathit{t}}^{\stackrel{\u203e}{\mathit{\alpha}}}{\mathit{x}}_{2},\dots ,{\mathit{t}}^{\stackrel{\u203e}{\mathit{\alpha}}}{\mathit{x}}_{\mathit{n}})$;

∙ If *T* is any motion of ${\mathbb{R}}^{\mathit{n}}$, then let ${\mathit{\gamma}}_{\mathit{x}}^{(\mathit{T})}$ be the line through $\mathit{x}=({\mathit{x}}_{1},\dots ,{\mathit{x}}_{\mathit{n}})\in {\mathbb{R}}^{\mathit{n}}$ defined as ${\mathit{\gamma}}_{\mathit{x}}^{(\mathit{T})}:=\mathit{T}\circ {\mathit{\gamma}}_{{\mathit{T}}^{-1}(\mathit{x})}$;

∙ Let ${\mathcal{H}}^{1}$ denote the one-dimensional Hausdorff measure in ${\mathbb{R}}^{\mathit{n}}$.

The main goal of this paper is to prove the following property: If ${\mathit{x}}_{0}$ is an *m*-density point of a Lebesgue measurable set *E* and *T* is an arbitrary motion of ${\mathbb{R}}^{\mathit{n}}$ mapping the origin to ${\mathit{x}}_{0}$, then we have

$$\underset{\mathit{t}\to 0+}{lim\hspace{0.17em}sup}\frac{{\mathcal{H}}^{1}(\mathit{E}\cap {\mathit{\gamma}}_{\mathit{x}}^{(\mathit{T})}((0,\mathit{t}]))}{{\mathcal{H}}^{1}({\mathit{\gamma}}_{\mathit{x}}^{(\mathit{T})}((0,\mathit{t}]))}=1$$

for almost every $\mathit{x}\in \mathit{T}(\{1\}\times {\mathbb{R}}^{\mathit{n}-1})$. An application of this result to locally finite perimeter sets is provided.