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

作者：Giannopoulos Apostolos; Hioni Labrini; Tsolomitis Antonis^*

来源：Advances in Applied Mathematics, 2016, 75: 116-143.

DOI：10.1016/j.aam.2016.01.004

• Isotropy
• Polytope
• Quermassintegral
• Covering number
• Projection
• Diameter

Let x(1),...,x(N) be independent random points distributed according to an isotropic log-concave measure mu on R-n, and consider the random polytope
K-N := conv{+/- x(1),...,x(N)}.
We provide sharp estimates for the quermassintegrals and other geometric parameters of K-N in the range cn <= N <= exp(n); these complement previous results from [13] and [14] that were given for the range cn <= N <= exp(root n). One of the basic new ingredients in our work is a recent result of E. Milman that determines the mean width of the centroid body Z(q)(mu) of mu for all 1 <= q <= n.

• 出版日期2016-4