This paper contains the result you desire, and many pointers to the (very abundant) literature no this topic.

Turns out your intuition is only close to the real answer, which in your notation is $$ V \simeq N^{\frac{n-1}{n+1}} $$ (see Section 9 of the above paper).

The current research is more focused about higher-order results (CLT etc.), but the Poissonian case (where the number $N$ is chosen randomly in precisely the way that gives independence to the events "there is a point in $A$" and "there is a point in $B$" whenever $A$ and $B$ are disjoint) is much better understood, as far as I know.

The result turns out to be very different depending whether you draw your points from a polytope or a smooth body.

This paper (Random Points and Lattice Points in Convex Bodies, Bárány in Bull. AMS 2008) contains the result you desire, and many pointers to the (very abundant) literature no this topic.

Turns out your intuition is far from the right answer, which as pointed out by Joseph O'Rourke is $$ V \simeq (\log N)^{n-1}$$ for points drawn in a polytope, but it is quite closer to the answer for points drawn from a smooth convex body: $$ V \simeq N^{\frac{n-1}{n+1}} $$ (see Section 9 of the above paper for both results).

The current research is more focused about higher-order results (CLT etc.), but the Poissonian case (where the number $N$ is chosen randomly in precisely the way that gives independence to the events "there is a point in $A$" and "there is a point in $B$" whenever $A$ and $B$ are disjoint) is much better understood, as far as I know.