Some porosity-type properties of sets related to the $d$-Hausdorff content

Abstract

Let $S\subset {\mathbb{R}}^n$ be a nonempty set. Given $d\in [0,n)$ and a cube $\overline{Q}\subset {\mathbb{R}}^n$ with $l=l(\overline{Q})\in (0,1]$, we show that if the $d$-Hausdorff content ${\mathcal{H}}_{\infty }^d(\overline{Q}\cap S)<\overline{\lambda }{l}^d$ for some $\overline{\lambda }\in (0,1)$, then the set $\overline{Q}\setminus S$ contains a specific cavity. More precisely, we prove existence of a pseudometric $\rho ={\rho }_{S,d}$ such that for each sufficiently small $\delta >0$ the $\delta$-neighborhood $U_{\delta l}^{\rho }(S)$ of $S$ in the pseudometric $\rho$ does not contain the whole $\overline{Q}$. Moreover, we establish the existence of constants $\overline{\delta }=\overline{\delta }(n,d,\overline{\lambda })>0$ and $\underline{\gamma }=\underline{\gamma }(n,d,\overline{\lambda })>0$ such that ${\mathcal{L}}^n(\overline{Q}\setminus U_{\delta l}^{\rho }(S))\ge \underline{\gamma }{l}^n$ for all $\delta \in (0,\overline{\delta })$. If, in addition, the set $S$ is $d$-lower content regular, we prove existence of a constant $\underline{\tau }=\underline{\tau }(n,d,\overline{\lambda })>0$ such that the cube $\overline{Q}$ is $\underline{\tau }$-porous. The sharpness of the results is illustrated by several examples.

arXiv commentMain theorems in the new version are more powerful. Proofs of the main results were polished