typical implementation of $(1+\varepsilon)$-coverings are the logarithmic $(1+\varepsilon)$-coverings, which consist in taking one vector of $R$ in each reciprocal image of $\bm{\varphi}(R)$. That is, for each $\bm{l}\in\bm{\varphi}(R)$, take one $\bm{\rho}$ in $\bm{\varphi}^{-1}(\bm{l})$. The logarithmic grid is depicted in Fig. \[fig:mo:approx\].

Now we must specify rigorously what approximate representations $E^{(\varepsilon_1)}$ of set ${\mbox{WST}}[{\mathcal{E}}]$, and $F^{(\varepsilon_2)}$ of set ${\mathcal{F}}$ we should take in input, in order to guarantee that $R[E^{(\varepsilon_1)},F^{(\varepsilon_2)}]$ is an $(1+\varepsilon)$-covering of $R[{\mbox{WST}}[{\mathcal{E}}],{\mathcal{F}}]$. Definitions \[def:under:covering\] and \[def:stick:covering\] come from the need of specific approximate representations that will carry the guarantees to the approximate final output $R[E^{(\varepsilon_1)},F^{(\varepsilon_2)}]$.

\[def:under:covering\] Given $\varepsilon>0$, $E\subset{\mathbb{R}^d_{+}}$ and $E^{(\varepsilon)}\subset{\mathbb{R}^d_{+}}$, $E^{(\varepsilon)}$ $(1+\varepsilon)$-[under]{}-covers $E$ if and only if: $$\begin{aligned}
\forall \bm{y}\in E,~~
\exists \bm{y}'\in E^{(\varepsilon)} &:&
\bm{y}{\succsim}\bm{y}'\\
\text{and}~~~
\forall \bm{y}'\in E^{(\varepsilon)},~~
\exists \bm{y}\in E &:&
(1+\varepsilon)\bm{y}'{\succsim}\bm{y}\end{aligned}$$

The first condition states that $E^{(\varepsilon)}$ bounds $E$ from below. The second condition states that this lower bound is precise within a multiplicative $(1+\varepsilon)$. Given $E$, one can implement Definition \[def:under:covering\] by using the log-grid (see e.g. Fig. \[fig:mo:approx\]): $$E^{(\varepsilon)}\leftarrow{\mbox{WST}}\left[~~\left\{~\bm{e}^{\bm{l}}\in{\mathbb{R}^d_{+}}\mid \bm{l}\in\bm{\varphi}\left({\mbox{WST}}[{\mathcal{E}}]\right)\right\}~~\right]$$ where $\bm{\varphi}({\mbox{WST}}[{\mathcal{E}}])=\{\bm{\varphi}(\bm{y})\in{\mathbb{N}^d}\mid\bm{y}\in {\mbox{WST}}[{\mathcal{E}}]\}$, and given $\bm{l}\in{\mathbb{N}^d}$, the vector $\bm{e}^{\bm{l}}$ is defined by $(\bm{e}^{\bm{l}})_k=(1+\varepsilon)^{l_k}$. Now let us state what approximation is required on the set of efficient outcomes ${\mathcal{F}}$.

\[def:stick:covering\] Given $\varepsilon>0$, $F\subset{\mathbb{R}^d_{+}}$ 