Asymptotic shape of the convex hull of isotropic log-concave random vectors
| Autor: | Apostolos Giannopoulos, Antonis Tsolomitis, Labrini Hioni |
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| Rok vydání: | 2016 |
| Předmět: |
Convex hull
Mathematics::Commutative Algebra Applied Mathematics 010102 general mathematics Isotropy Centroid Metric Geometry (math.MG) Polytope Covering number 01 natural sciences Measure (mathematics) Primary 52A21 Secondary 46B07 52A40 60D05 010101 applied mathematics Combinatorics Mathematics - Metric Geometry FOS: Mathematics 0101 mathematics Mathematics Complement (set theory) Mean width |
| Zdroj: | Advances in Applied Mathematics. 75:116-143 |
| ISSN: | 0196-8858 |
| DOI: | 10.1016/j.aam.2016.01.004 |
| Popis: | Let $x_1,\ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$, and consider the random polytope $$K_N:={\rm conv}\{ \pm x_1,\ldots ,\pm x_N\}.$$ We provide sharp estimates for the querma\ss{}integrals and other geometric parameters of $K_N$ in the range $cn\ls N\ls\exp (n)$; these complement previous results from \cite{DGT1} and \cite{DGT} that were given for the range $cn\ls N\ls\exp (\sqrt{n})$. 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 $1\ls q\ls n$. |
| Databáze: | OpenAIRE |
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