TY - JOUR

T1 - Random polytopes with vertices on the boundary of a convex body

AU - Scḧtt, Carsten

AU - Werner, Elisabeth

PY - 2000/11/1

Y1 - 2000/11/1

N2 - Let K be a convex body in ℝn 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 ℙf 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 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 density (formula presented)

AB - Let K be a convex body in ℝn 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 ℙf 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 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 density (formula presented)

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U2 - 10.1016/s0764-4442(00)01685-2

DO - 10.1016/s0764-4442(00)01685-2

M3 - Article

AN - SCOPUS:0034311783

SN - 0764-4442

VL - 331

SP - 697

EP - 701

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

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

IS - 9

ER -