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$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

For the standard cube
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$c\u2254\mathrm{ExampleSets}:\mathrm{Cube}\left(\right)$

${c}{\u2254}{\{}\begin{array}{lll}{\mathrm{Coordinates}}& {\:}& \left[{{x}}_{{1}}{\,}{{x}}_{{2}}{\,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {\:}& \left[{}{{x}}_{{3}}{\le}{1}{\,}{{x}}_{{3}}{\le}{1}{\,}{}{{x}}_{{2}}{\le}{1}{\,}{{x}}_{{2}}{\le}{1}{\,}{}{{x}}_{{1}}{\le}{1}{\,}{{x}}_{{1}}{\le}{1}\right]\end{array}$
 (1) 
the edges of the cube all have length 2
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$\mathrm{c\_edges}\u2254\mathrm{Edges}\left(c\right)\:$$\mathrm{map}\left(\mathrm{Length}\,\mathrm{c\_edges}\right)$

$\left[{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}{\,}{2}\right]$
 (2) 
the cube has a volume of 8
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$\mathrm{Volume}\left(c\right)$

and the total surface area of the cube is
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$\mathrm{Area}\left(c\right)$

The polyhedral set ps is a threedimensional polytope in fourdimensional space.
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$\mathrm{ps}\u2254\mathrm{PolyhedralSet}\left(\left[\left[1\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\,\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,1\right]\right]\,\left[x\,y\,z\,u\right]\right)$

${\mathrm{ps}}{\u2254}{\{}\begin{array}{lll}{\mathrm{Coordinates}}& {\:}& \left[{x}{\,}{y}{\,}{z}{\,}{u}\right]\\ {\mathrm{Relations}}& {\:}& \left[{}{z}{\le}{0}{\,}{}{y}{\le}{0}{\,}{}{x}{\le}{0}{\,}{x}{+}{y}{+}{z}{\le}{1}{\,}{x}{+}{y}{+}{z}{+}{u}{=}{1}\right]\end{array}$
 (5) 
Its fourdimensional volume is 0, but its threedimensional volume is positive.
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$\mathrm{Volume}\left(\mathrm{ps}\right)$

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$\mathrm{Volume}\left(\mathrm{ps}\,3\right)$
