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 $\mathcal{H}^{d}_{\infty}(\overline{Q} \cap S) < \overline{\lambda}l^{d}$ 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 $U^{\rho}_{\delta l}(S)$ 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 $\mathcal{L}^{n}(\overline{Q} \setminus U^{\rho}_{\delta l}(S)) \geq \underline{\gamma} l^{n}$ 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.