Let $d$ be an integer. I will work in $\mathbb{R}^d$.
Let $N$ be an integer and $x_0,x_1,\cdots x_N$ be $N+1$ points of $\mathbb{R^d}$.
If $N=d$, then I have $d+1$ points and $Vol(Conv(x_0,x_1,\cdots,x_d))=\frac{det(x_1-x_0,x_2-x_0,\cdots x_d-x_0)}{d!}$ because we can view it as a linear transformation of the simplex which is of measure $1/d!$.
However is there the same kind of formula if $n$ is strictly bigger than $d$? Or at least useful bounds on the volume of $Conv(x_0,x_1,\cdots x_n)$ when $N$ is big?
Of course the formula above can not be true anymore because it will just be zero.
The volume of the convex set is no greater than the sum of the volumes of the simplices spanned by $N+1$-tuples of points, so the sum of determinants will be a bound. How useful it is depends on what you want to use it for...