n be the discrete cube in R n . For every N ≥ n we consider the class of convex bodies K N = co{±x 1 , . . . , ±x N } which are generated by N random points x 1 , . . . , x N chosen independently and uniformly from E n 2 . We show that if n ≥ n 0 and N ≥ n(log n) 2 , then, for a random K N , the inradius, the volume radius, the mean width and the size of the maximal inscribed cube can be determined up to an absolute constant as functions of n and N . This geometric description of K N leads to sharp estimates for several asymptotic parameters of the corresponding n-dimensional normed space X N .