I have a polytope $P$ described as the convex hull of finite points $u_1,..., u_m\in \mathbb R^n$. Is there an easy way to compute the volume of $P$ in $\mathbb R^n$?
So far I have it written as $$ P = \{ Ux: 0\leq x\leq 1, x^T\mathbf 1 = 1\} $$ where the columns of $U$ are $u_i$. I could conceivably get this into the form $$ P = \{x : Ax \leq b\} $$ for a clever choice of $A$ and $b$.
Any slick ways of computing the volume of $P$, using either $U$ or $A$ and $b$?
In reference to this MathOverflow thread, it looks like the answer is "there is no easy, general way to find such a volume." If you need the volume of a specific polytope, the responses contain a link to Qhull software. For more about computations of polytope computation see Dr. Fukuda's FAQ and for more about the specifics of algorithms see this study.