2018/07/06 by Joscha Prochno, Prochno, Joscha, Christoph Thäle +3
Environmental Science · Mathematics · #46B06 #52A22 #52A23 #52B11 #60D05 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Heavy metals in environment #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #Toxic Organic Pollutants Impact
paper · pdf · doi:10.48550/arxiv.1807.02396
openalex publication_date 2018/07/06 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
For an isotropic convex body K\⊂\ℝn we consider the isotropic\nconstant LKN of the symmetric random polytope KN generated by N\nindependent random points which are distributed according to the cone\nprobability measure on the boundary of K. We show that with overwhelming\nprobability LKN\≤ C\√(\log(2N/n)), where C\∈(0,\∞) is an\nabsolute constant. If K is unconditional we argue that even LKN\≤ C\nwith overwhelming probability. The proofs are based on concentration\ninequalities for sums of sub-exponential or sub-Gaussian random variables,\nrespectively, and, in the unconditional case, on a new \ψ2-estimate for\nlinear functionals with respect to the cone measure in the spirit of Bobkov and\nNazarov, which might be of independent interest.\n