For $s,t\in {\mathbb{R}}$, we write $s\wedge t : = \min\{ s,t\}$ and $s \vee t := \max\{s,t\}$. We denote the $d$-dimensional Euclidean space by ${\mathbb{R}^d}$, ${\mathbb{N}}$ is the set of natural numbers and ${\mathbb{N}}^* := {\mathbb{N}}\setminus \{ 0 \}$. For each $x,y \in {\mathbb{R}^d}$, $|x|$ denotes the Euclidean length of $x$ and $x\cdot y$ the scalar product. For $r>0$, we set $B_r(x): = \{ y\in {\mathbb{R}^d}: |x-y| < r\}$ and $B_r : = B_r(0)$. If $K$ is a subset of ${\mathbb{R}}^d$, we denote by $\overline K$ and $\partial K$ its closure and its boundary and let $K+B_r$ be the set of points which are at a distance at most $r$ of $K$. The Hausdorff distance between two subsets $U,V\subseteq {\mathbb{R}^d}$ is $\operatorname{dist}_H(U,V)=\inf\{r\geq 0 : U\subseteq V+r B_1\ {\rm and }\ V\subseteq U+r B_1\}$. The set of bounded and Lipschitz continuous maps on ${\mathbb{R}^d}$ is denoted $W^{1,\infty}({\mathbb{R}}^d)$. If $E$ is a set, then ${\mathds{1}}_E$ is the indicator function of $E$. The denote the set of upper and lower semicontinuous functions on a domain $E\subseteq{\mathbb{R}^d}$ by $\operatorname{USC}(E)$ and $\operatorname{LSC}(E)$, respectively. The space of bounded and uniformly continuous functions is denoted $\operatorname{BUC}(E)$.

We usually [*do not*]{} display the dependence of the various quantities 