Abstract
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)
| Translated title of the contribution | Random polytopes with vertices on the boundary of a convex body |
|---|---|
| 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 |