## Abstract

Let K be a convex body in ℝ^{n} and let f : ∂K → ℝ+ be a continuous, positive function with ∫_{∂K} f (x) dμ∂K (X) = 1, where μ∂K is the surface measure on ∂K. Let ℙ_{f} be the probability measure on ∂K given by dℙ _{f} (x) = f(x) dμ∂K (x). Let K be the (generalized) Gauß-Kronecker curvature and E(f, N) the expected volume of the convex hull of N points chosen randomly on ∂K with respect to ℙ _{f}. Then, under some regularity conditions on the boundary of K, (formula presented) where c_{n} is a constant depending on the dimension n only. The minimum at the right-hand side is attained for the normalized affine surface area measure with density (formula presented)

Translated title of the contribution | Random polytopes with vertices on the boundary of a convex body |
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Original language | French |

Pages (from-to) | 697-701 |

Number of pages | 5 |

Journal | Comptes Rendus de l'Academie des Sciences - Series I: Mathematics |

Volume | 331 |

Issue number | 9 |

State | Published - Nov 1 2000 |