Quermaßintegrals and asymptotic shape of random polytopes in an isotropic convex body

Affine quermassintegrals of random polytopes

“…For a proof of all these assertions see [12], [13], and also [9,Chapter 11]. Moreover, in the range n N e √ n , one can further check that an upper bound of the order √ log N L K holds for the volume radius of a random k-dimensional projection of a random K N (see [13,Fact 4.6]). Starting from the inequality…”

“…, ±x N }. We provide a description of the "asymptotic shape" of M N which is parallel to the available description for K N ; this can be done with suitable modifications of the theory developed in [12] and [13]. This allows us to prove the analogue of Theorem 1.2 for this model too.…”

Asymptotic shape of the convex hull of isotropic log-concave random vectors

“…Second, one has (1.10) |K N | 1/n c 7 log(2N/n) √ n with probability greater than 1 − 1 N , where C > 0 is an absolute constant. In [14] these results were extended to the full family of quermaßintegrals W n−k (K N ) of K N . These are defined through Steiner's formula…”

“…The study of the asymptotic shape of random polytopes whose vertices have a log-concave distribution was initiated in [13] and [14]. Given an isotropic log-concave measure µ on R n , for every N n we consider N independent random points x 1 , .…”

“…A more careful analysis (which can be found in [14,Theorem 1.2]) shows that if n 2 N exp( √ n) then, for any s 1, a random K N satisfies, with probability greater than 1 − N −s ,…”