New analysis of the sphere covering problems and optimal polytope approximation of convex bodies

In this paper, we show that both sphere covering problems and optimal polytope approximation of convex bodies are related to optimal Delaunay triangulations, which are the triangulations minimizing the interpolation error between function ∥x∥2 and its linear interpolant based on the underline triangulations. We then develop a new analysis based on the estimate of the interpolation error to get the Coxeter-Few-Rogers lower bound for the thickness in the sphere covering problem and a new estimate of the constant deln appeared in the optimal polytope approximation of convex bodies.