that for each $y$ $S_h(y)$ is a closed convex polytope (and in case $S_h(y)\subset{\mathcal{B}}$, $S_h(y)$ is compact), and the identity .

\[remcompact\] $S_h(y)$ defined by is also a *unbounded convex set*, where only the boundary points take the unbounded values, see Figure \[fig:Sh\]. However, $S_h(y)\subset{\mathcal{B}}$ defined as $S_h(y) := \left\{ z\in{\mathcal{B}}\colon \text{$z$ is $(y,h)$-monotone} \right\}$ is obviously a compact convex set. Analogously, in the unbounded case $C(h)\subset{\mathbb{R}}^N$ is a finite union of polytopes, and $C(h)\subset{\mathcal{B}}$ is additionally compact, hence a simplicial set.

All elements of the proof of Theorem \[thm:main\] are the same regardless if we consider the polytope sets $S_h(y)$ being unbounded (over ${\mathbb{R}}^N$) or compact (over ${\mathcal{B}}$).

(0,0) – (0,5); (0,0) – (8,0);

(-0.1,0.5) – (0,0.5); (-0.1,1) – (0,1); (-0.1,1.5) – (0,1.5); (-0.1,2) – (0,2); (-0.1,2.5) – (0,2.5); (-0.1,3) – (0,3); (-0.1,3.5) – (0,3.5);

(1,-0.1) – (1,0.1); (2,-0.1) – (2,0.1); (3,-0.1) – (3,0.1); (4,-0.1) – (4,0.1); (5,-0.1) – (5,0.1); (6,-0.1) – (6,0.1); (7,-0.1) – (7,0.1);

(2,0.5) circle (2pt); (4,3.5) circle (2pt); (6,2) circle (2pt);

(1,0.5) – (1,5.5); (3,0.5) – (3,3.5); (5,2) – (5,3.5); (7,2) – (7,5.5);

(1,-0.3) node [$z_1$]{}; (2,-0.3) node [$z_2$]{}; (3,-0.3) node [$z_3$]{}; (4,-0.3) node [$z_4$]{}; (5,-0.3) node [$z_5$]{}; (6,-0.3) node [$z_6$]{}; (7,-0.3) node [$z_7$]{};

(-0.3,1) 