I would like to give a definition of polyhedral space in $\mathbb{R}^n$ that is easy to understand by people that has some Maths knowledge but are neither expert in Calculus nor any other Mathematics field.
Actually, I am not sure about the right concept for polyhedral space. Is it the same as convex polyhedral set? Is it a finite union of convex polyhedral sets?
EDIT: I ask this question here because I haven't been able to find an answer by myself. Particularly, I haven't been able to understand the concept of polyhedral space.
EDIT: Well, I've been surfing a little bit longer, and I've found a definition that I understand.
$X \subseteq \mathbb{R}^n$ is a polyhedral space (or set) if there is a matrix $\mathbf{A}$ and a vector $\mathbf{b}$ such that $X=\left\{ \mathbf{x} \in \mathbb{R}^n |\mathbf{Ax} \leq \mathbf{b} \right\}$.
Is this right? This is the same as what I call a convex polyhedral set.
This is a reasonable definition. It might not be exactly the one you want: you might want to permit finite unions. If you're willing to take finite unions, you no longer keep convexity, so you might be interested in a more general, intrinsic, notion: most likely, that of simplicial complexes. Simplicial complexes are almost exactly the same as unions of convex polyhedra, except that they don't come embedded in Euclidean space and you make sure that the convex pieces line up nicely.
EDIT: The definition of polyhedral space one finds on Wikipedia is not the same as the one you've given. Wikipedia permits any simplicial complex with a metric flat on each simplex, which is essentially the same thing as a union, not necessarily finite, of polyhedral spaces under your definition which only intersect along faces.