## On the expected number of shadow vertices of the convex hull of random points

- Let \(a_1,\dots,a_m\) be independent random points in \(\mathbb{R}^n\) that are independent and identically distributed spherically symmetrical in \(\mathbb{R}^n\). Moreover, let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_m\) and let \(L_k\) be an arbitrary \(k\)-dimensional subspace of \(\mathbb{R}^n\) with \(2\le k\le n-1\). Let \(X_k\) be the orthogonal projection image of \(X\) in \(L_k\). We call those vertices of \(X\), whose projection images in \(L_k\) are vertices of \(X_k\)as well shadow vertices of \(X\) with respect to the subspace \(L_k\) . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of \(X\) in \(L_k\).

Author: | Karl-Heinz Küfer |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-50516 |

Serie (Series number): | Preprints (rote Reihe) des Fachbereich Mathematik (282) |

Document Type: | Report |

Language of publication: | English |

Publication Date: | 2017/11/09 |

Year of Publication: | 1996 |

Publishing Institute: | Technische Universität Kaiserslautern |

Date of the Publication (Server): | 2017/11/09 |

Number of page: | 15 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 510 Mathematik |

Licence (German): | Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0) |