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

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

$$f_{as}(x ) = \frac{\kappa(x)^{\frac{1}{n + 1}}}{\int_{\partial K}\kappa(x)^{\frac{1}{n + 1}}{\rm d}\mu_{\partial K}(x)}. $$

Carsten Schütt: Partially supported by the Schrödinger Institute, Vienna, and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, Jerusalem, sponsored by the Minerva Foundation (Germany).

Elisabeth Werner: Partially supported by the Schrödinger Institute, Vienna, by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, Jerusalem, sponsored by the Minerva Foundation (Germany) and by a grant from the National Science Foundation.