For a given $(\lambda,\nu)$ we define subsets $U_{v}^{\beta}\subset U_{v}$ and $V_{w}^{\beta^{\vee}}\subset V_{w}$ whose union will give the complement of a neighborhood of $D$ depending on two real parameters $\beta,{\beta^{\vee}}>0$. We assume $\beta,{\beta^{\vee}}$ to be small in the $\lambda,\nu$ scales, respectively. In what follows we identify $\Sigma$ with $\partial\Delta^{\vee}_{\lambda}$ and $\partial{\Delta_{\nu}}$ via the maps $\phi$ and $\hat{\phi}$ associated with the bi-PIKAS of type $(\lambda,\nu)$.

For $v\in\operatorname{vert}(S)$ we let $U_{v}^{\beta}$ be the set of points in the corresponding facet of $\Delta^{\vee}_{\lambda}$ which lie in the closed polyhedron $Q^{\lambda}_{(v\mid\{0\})}(\beta)$ (cf. [HZ02, Section 3.2]), and similar for $V_{w}^{\beta^{\vee}}$. Explicitly,

	$$\displaystyle U_{v}^{\beta}:=\{n\in U_{v}\ :\ \langle m,n\rangle+\lambda(m)\leq-\beta,\text{ all }m\in\Delta_{\mathbb{Z}}\backslash\{v,0\}\},$$
	$$\displaystyle V_{w}^{\beta^{\vee}}:=\{m\in V_{w}\ :\ \langle m,n\rangle+\nu(n)\leq-{\beta^{\vee}},\text{ all }n\in\Delta^{\vee}_{\mathbb{Z}}\backslash\{w,0\}\}.$$

Because $\beta,\beta^{\vee}$ are small the sets $U_{v}^{\beta}$ and $V_{w}^{\beta^{\vee}}$ are non-empty. We define the smooth part of $\Sigma$ as

$$\Sigma^{\mathrm{sm}}:=\bigcup_{v\in\operatorname{vert}(S)}U_{v}^{\beta}\cup\bigcup_{w\in\operatorname{vert}(T)}V_{w}^{\beta^{\vee}}.$$

Then the neighborhood of $D$ is defined as the complement to all these closed sets in $\Sigma$:

$$N_{\lambda,\nu}^{\beta,\beta^{\vee}}(D):=\Sigma\backslash\Sigma^{\mathrm{sm}}.$$

An important observation is that $N_{\lambda,\nu}^{\beta,\beta^{\vee}}(D)\to D$ as $\beta,\beta^{\vee}\to 0$ in the scales of $\lambda,\nu$, respectively.