We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach–Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case.
We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.
"A multivariate Gnedenko law of large numbers." Ann. Probab. 41 (5) 3051 - 3080, September 2013. https://doi.org/10.1214/12-AOP804