A Fine Property of Sets at Points of Lebesgue Density

Abstract

Consider an integer $\mathit{n}\ge 2$, $\mathit{m}\in [\mathit{n},+\infty )$ and put $\bar{\mathit{\alpha }}:=\frac{\mathit{m}-1}{\mathit{n}-1}$. 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 ${\mathit{\gamma }}_{\mathit{x}}(\mathit{t}):=(\mathit{t}{\mathit{x}}_1,{\mathit{t}}^{\bar{\mathit{\alpha }}}{\mathit{x}}_2,\dots ,{\mathit{t}}^{\bar{\mathit{\alpha }}}{\mathit{x}}_{\mathit{n}})$;

∙ 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.