Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body

  • Carsten SchüttEmail author
  • Elisabeth Werner
Part of the Lecture Notes in Mathematics book series (LNM, volume 1807)


Let K be a convex body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gauß-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K
$$ \lim_{ N\to \infty} \frac{{\rm vol}_n (K ) -\mathbb{E} (f,N )}{\left(\frac{1}{N}\right)^{\frac{2}{n-1}}} = c_n\int_{\partial K}\frac{\kappa(x)^{\frac{1}{n-1}}}{f(x)^{\frac{2}{n-1}}} {\rm d}\mu_{\partial K}(x),$$
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
$$f_{as}(x ) = \frac{\kappa(x)^{\frac{1}{n + 1}}}{\int_{\partial K}\kappa(x)^{\frac{1}{n + 1}}{\rm d}\mu_{\partial K}(x)}. $$

Mathematics Subject Classification (2000):

46-06 46B07 52-06 60-06 


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Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian Albrechts UniversitätKielGermany
  2. 2.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  3. 3.Ufr de MathématiqueUniversité de Lille 1Villeneuve d’AscqFrance

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