Approximation of smooth convex bodies by random polytopes

作者：Grote Julian^*; Werner Elisabeth

来源：Electronic Journal of Probability, 2018, 23(0): 9.

DOI：10.1214/17-EJP131

• random polytopes
• approximation
• convex bodies

Let K be a convex body in R-n and f : partial derivative K -> R+ a continuous, strictly positive function with integral(partial derivative K) f (x) d mu(partial derivative K) (x) = 1. We give an upper bound for the approximation of K in the symmetric difference metric by an arbitrarily positioned polytope P-f in R-n having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner [36]. The polytope P-f is obtained by a random construction via a probability measure with density f. In our result, the dependence on the number of vertices is optimal. With the optimal density f, the dependence on K in our result is also optimal.

• 出版日期2018