Some Porosity-Type Properties of Sets Related to the $d$-Hausdorff Content

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 side length $l=l(\overline {Q}) \in (0,1]$, we show that if the $d$-Hausdorff content $\mathcal H^d_{\infty }(\overline {Q}\cap S)$ of the set $\overline {Q}\cap S$ satisfies the inequality $\mathcal H^d_{\infty }(\overline {Q}\cap S)<\overline {\lambda }l^{d}$ for some $\overline {\lambda }\in (0,1)$, then the set $\overline {Q}\setminus S$ contains a specific cavity. More precisely, we prove the existence of a pseudometric $\rho =\rho _{S,d}$ such that for every sufficiently small $\delta >0$ the $\delta $-neighborhood $U^\rho _{\delta _{}}(S)$ of $S$ in the pseudometric $\rho $ does not cover $\overline {Q}$. 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 $\delta \in (0,\overline {\delta })$, where $\mathcal L^n$ is the Lebesgue measure. If in addition the set $S$ is lower content $d$-regular, we prove the 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.