Volume of a simplex by using LP or MIP

Question

Is there any way to calculate the volume of a simplex by using LP or MIP when we have its extreme points? Is there any paper that gives me some clues?

Answer 1

If it really is just a simplex (the convex hull of $n+1$ points in $\mathbb R^n$) then the determinant formula for the area of a simplex generalizes to give us $$V = \frac{1}{n!} \left| \det \begin{bmatrix} 1 & x_{11} & x_{12} & \dots & x_{1n} \\ 1 & x_{21} & x_{22} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n+1,1} & x_{n+1,2} & \dots & x_{n+1,n}\end{bmatrix}\right|$$ where $(x_{11},\dots, x_{1n}), (x_{21}, \dots, x_{2n}), \dots, (x_{n+1,1}, \dots, x_{n+1,n})$ are the coordinates of your extreme points.