Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_N$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_N:=vol\left(K_N\cap RB₂ⁿ\right)/vol\left(RB₂ⁿ\right)$. For a large range of R = R(n), we establish a sharp threshold for N, above which $V_N\to 1$ as n → ∞, and below which $V_N\to 0$ as n → ∞. We also consider the case when $K_N$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0,1) and...