Suppose $X_1, ... , X_{n+1}$ are $n$ i.d.d. random points in $\mathbb{R}^n$ with distribution $N(0, I)$ (multivariate normal with zero mean and identity covariance matrix). Suppose $C$ is the convex hull of $X_1, … , X_{n+1}$. It is not hard to see, that $C$ is almost surely an $n$-dimensional simplex. My question is - what the asymptotic of it’s expected $n$-dimensional Lebesgue measure as $n \to \infty$?
Note, that the explicit computation of the values of that sequence seems to be quite hard in general:
For $n = 1$ (the easiest case) the expectation is $\frac{2}{\sqrt{\pi}$. For $n = 2$ it can be represented as an integral using Heron’s formula, but I am too lazy to explicitly calculate it. How to obtain the exact numbers for higher dimensions, is a complete mystery to me…
However, I am interested not in exact values, but in asymptotic for $n \to \infty$ (and, unfortunately, currently have no idea how to get it).
