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

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

## Abstract

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