Hence, by Proposition \[proposition-Near-sel-v8:1\], we get that $h_{\sigma_1}\operatorname{\upharpoonright}|\tau|= h_\tau=h_{\sigma_2}\operatorname{\upharpoonright}|\tau|$. Accordingly, we may define a map $h:|\Sigma|\to Y$ by $h\operatorname{\upharpoonright}|\sigma|=h_\sigma$, $\sigma\in \Sigma$. This map $h$ is as required. Indeed, it is continuous because so is each $h_\sigma$, $\sigma\in \Sigma$. To see that $h$ also has the property in , take a simplex $\sigma\in \Sigma$, and set $v=\min\sigma$. If $\sigma=\{v\}$, then by \[item:Near-sel-v8:1\] of Proposition \[proposition-Near-sel-v8:1\], $h(|\sigma|)=h(v)=h_\sigma(v)=p_v\in \Psi(v)$. Otherwise, if $\tau=\sigma\setminus\{v\}\neq {\varnothing}$, it follows from \[item:Near-sel-v8:2\] of Proposition \[proposition-Near-sel-v8:1\] that for every $z\in |\tau|$ and $t\in [0,1]$, $$h\big(tv+(1-t)z\big)= h_\sigma\big(tv
    +(1-t)z\big)=H_v(h_\sigma(z),t)\in \Psi(v).$$ Accordingly, $h(|\sigma|)\subset \Psi(v)$ because $|\sigma|=|\tau|*v$.

Recall that the $k$-skeleton $\Sigma^k$, $k\geq 0$, of a simplicial complex $\Sigma$ is the simplicial subcomplex $\Sigma^{k}=\{\sigma\in \Sigma:\operatorname{Card}(\sigma)\leq k+1\}$. A simplicial complex $\Sigma$ is called *finite-dimensional* if $\Sigma=\Sigma^k$ for some ${k\geq 0}$, and we say that $\Sigma$ is *$k$-dimensional* if $k\geq0$ is the smallest integer with this property, i.e. for which $\Sigma=\Sigma^k$. If $\Sigma$ is an $(n+1)$-dimensional simplicial complex which is oriented with respect to some partial order $\leq$ on $\Sigma^0$, then we may consider the chain of subcomplexes $\Sigma_{n+1}\subset \Sigma_n\subset \dots \subset \Sigma_0=\Sigma$ defined by $$\label{eq:Near-sel-v9:2}
  \begin{cases}
    \Sigma_k=(\Sigma_k)^{n+1-k}, &\text{and}\\
    \min\sigma\notin (\Sigma_{k+1})^0, &\text{whenever
      $\sigma\in \Sigma_k\setminus \Sigma_{k+1}$.}
  \end{cases}$$ Namely, let $V_0$ be the minimal elements of $\Sigma^0$, 