Covering functionals of convex polytopes with few vertices

By using elementary yet interesting observations and refining techniques used in a recent work by Fei Xue et al., we present new upper bounds for covering functionals of convex polytopes in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} with few vertices. In these estimations, no information other than the number of vertices of the convex polytope is used.