Smallest singular value of random matrices and geometry of random polytopes

We study the behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then, we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes (absolutely convex hulls of rows of random matrices). All our results hold with high probability, that is, with probability exponentially (in dimension) close to 1.