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- W1965702936 abstract "Let K be a convex body in Rn and let f:∂K→R+ be a continuous, positive function with ∫∂Kf(x)dμ∂K(x)=1, where μ∂K is the surface measure on ∂K. Let Pf be the probability measure on ∂K given by dPf(x)=f(x)dμ∂K(x). Let κ 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 Pf. Then, under some regularity conditions on the boundary of K,limN→∞voln(K)−E(f,N)(1N)2n−1=cn∫∂Kκ(x)1n−1f(x)2n−1dμ∂K(x),where cn 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 densityfas(x)=κ(x)1n+1∫∂Kκ(x)1n+1dμ∂K(x). Soit K un corps convexe dans Rn et soit f:∂K→R+ une fonction continue positive telle que ∫∂Kf(x)dμ∂K(x)=1, où μ∂K est la mesure de surface sur ∂K. Soit Pf la mesure de probabilité sur ∂K définie par dPf(x)=f(x)dμ∂K(x). Soient κ la courbure de Gauß–Kronecker (généralisée) et E(f,N) l'espérance du volume de l'enveloppe convexe de N points choisis aléatoirement sur ∂K par rapport à Pf. Alors on a sous certaines conditions de régularité de ∂K,limN→∞voln(K)−E(f,N)(1N)2n−1=cn∫∂Kκ(x)1n−1f(x)2n−1dμ∂K(x),où cn est une constante qui ne dépend que de la dimension n. Le minimum du membre de droite est atteint pour la mesure normalisée de surface affine ayant pour densitéfas(x)=κ(x)1n+1∫∂Kκ(x)1n+1dμ∂K(x)." @default.
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- W1965702936 date "2000-11-01" @default.
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- W1965702936 title "Random polytopes with vertices on the boundary of a convex body" @default.
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- W1965702936 doi "https://doi.org/10.1016/s0764-4442(00)01685-2" @default.
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