we consider a compact or noncompact version of $S_h$ (the application in Section \[secapplication\] is the sole exception), we do not provide distinct symbols for these two objects.

Let $n=3$, $N=7$, $h = (0.5,3.5,2)$, and $y=(2,4,6)$. The possible coordinate values of $z\in S_{h,N,n}(y)\subset {\mathbb{R}}^7$ are shown in Figure \[fig:Sh\]. Observe that $S_{h,N,n}(y)$ is an unbounded convex set.

Let us relate a preimage component $C(h)$ with the definition of $(y,h)$-monotonicity. First, observe that

\[eqCh2\] $$\begin{aligned}
  C(h) &= \left\{ z\in{\mathbb{R}}^{N}\colon \text{$z$ is $(y,h)$-monotone for any $y\in L_n^N$} \right\},\text{ and}\\
  C(h) &= \bigcup_{y\in L_n^N}{S_h(y)}.\label{Chsum}\end{aligned}$$ We define analogously the compact version of $C(h)\colon L_n^N \to 2^{{\mathcal{B}}}$, using compact $S_h$’s.

\[prop:Spolytope\] Given $h\in {{\mathcal{E}}}^n_+$, and $y\in L_n^N$. The set $S_h(y) = S_{h,N,n}(y)$ defined by is a *closed convex polytope*. $S_h(y)\subset{\mathcal{B}}$ defined as is additionally *compact*.

From the definition $S_h(y) = S_{h,N,n}(y)$ is the intersection of a finite number of half-spaces $z_1\geq z_2$, $z_2 \geq z_3,\dots,z_{y_1-1}\geq z_{y_1}=h_1$ etc.. Therefore $S_{h,N,n}(y)$ is a closed convex polytope. In case $z\in{\mathcal{B}}$ it is additionally compact, as the boundary points $z_1,z_N$ have restricted values.

\[prop:Cpolytope\] Given $h\in {{\mathcal{E}}}^n_+$. The set $C(h)$ is a finite union of polytopes. $C(h)\subset {\mathcal{B}}$ is a compact simplicial set.

Follows directly from Prop \[prop:Spolytope\] showing 