Polytope parametrization How one could parametrize a convex polytope?
By parametrization I mean something like in multiple integrals, when to integrate over an area one can integrate over one variable in an interval $[l,r]$ and inside over other in $[l(x),r(x)]$
More formally, how one could take a convex polytope (in H- or V-representation, does not matter to me) and describe it in H-representation as
$$
l_1 \le x_1 \le r_1 \\
l_2(x_1) \le x_2 \le r_2(x_1) \\
\ldots \\
l_d(x_1,\ldots,x_{d-1}) \le x_d \le r_d(x_1,\ldots,x_{d-1})
$$
Since not every polytope can be represented in that way, I'm also interested in a way to split a polytope to many that can.

It's hard: computing the volume of a convex polytope is $\#P$ hard. For various related issues , heuristics, and occasional algorithms, see this nice presentation by Jesus de Loera.
