Random polytopes with vertices on the boundary of a convex body

Carsten Scḧtt, Elisabeth Werner

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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)

Original languageEnglish (US)
Pages (from-to)697-701
Number of pages5
JournalComptes Rendus de l'Academie des Sciences - Series I: Mathematics
Issue number9
StatePublished - Nov 1 2000
Externally publishedYes


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