Consider the convex polytope $\mathcal{P} \subset \mathbb{R}^n$ with $\mathcal{H}$-representation
$$\mathcal{P} = \{x \in \mathbb{R}^n\colon Ax \leq b\},$$
where $A \in \mathbb{R}^{m\times n}$ and $b \in \mathbb{R}^m$. How to compute the volume of $\mathcal{P}$? (Can I convert this problem into an optimization problem?) Thanks!
In general, this is not possible in polynomial time (unless P=NP). You can do it in exponential time by so called triangulation, that is by dissecting your polytope into simplices whose volume can be computed using a closed formula.
Have a look here: https://doi.org/10.1007/978-94-011-0924-6_17