of *strictly ordered multi-indices* is given by $$L_m^N := \left\{(y_1,\dots,y_m)\in\{1,2,\dots,N\}^m\colon y_j < y_{j+1}\text{ for }1 \leq j < m\right\}.$$

\[defmonotone\] Let $h\in {{\mathcal{E}}}^n_+$ and $y\in L_n^N$. A point $z\in{\mathbb{R}}^N$ is *$(y,h)$-monotone* if the following conditions are satisfied.

1.  $z(y_k)=h_k$ for $k=1,\ldots,n$.

2.  $z$ is monotone increasing on ${\left\{ {1,\ldots,y_1}\right\}}$ and monotone decreasing on ${\left\{ {y_n,\ldots,N}\right\}}$.

3.  If $m$ is odd, then $z$ is monotone increasing on ${\left\{ {y_m,y_{m+1}}\right\}}$.

4.  If $m$ is even, $z$ is monotone decreasing on ${\left\{ {y_m,y_{m+1}}\right\}}$.

Observe that if $z$ is $(y,h)$-monotone, then $\operatorname{cv}(z) = h$.

If $z$ is as in Example \[ex:cv(z)\], then $z$ is $(y,\operatorname{cv}(z))$-monotone for $y=(2,5,6)$ or $y = (3,5,6)$.

Given $h\in {{\mathcal{E}}}^n_+$ define $S_h = S_{h,N,n}\colon L_n^N \to 2^{{\mathbb{R}}^{N}}$ by $$\label{slprop}
S_h(y) = S_{h,N,n} (y) := \left\{ z\in{\mathbb{R}}^{N}\colon \text{$z$ is $(y,h)$-monotone} \right\}.$$ Let $$\bar{a},\bar{b}>0$$ denote global bounds imposed on the first and the last components respectively and denote the subset of ${\mathbb{R}}^N$ having boundary elements bounded by $$\label{compactB}
{\mathcal{B}}= {\mathcal{B}}(\bar{a},\bar{b}) = \left\{z\in{\mathbb{R}}^N\colon z_1\in[-\bar{a},\bar{a}],\ z_N\in[-\bar{b},\bar{b}]\right\}.$$ Let $h\in {{\mathcal{E}}}^n_+$, define the compact version of $S_h = S_{h,N,n}\colon L_n^N \to 2^{{\mathcal{B}}}$ by $$\label{slpropcompact}
S_h(y) = S_{h,N,n} (y) := \left\{ z\in{\mathcal{B}}\colon \text{$z$ is $(y,h)$-monotone} \right\}.$$ Because most of the arguments are precisely the same regardless if 