Shortest path avoiding balls

Abstract

It is shown that in a packing of open circular discs with radii not exceeding 1, any two points lying outside the circles at distance d from one another can be connected by a path traveling outside the circles and having length at most \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{4} {\pi }d + O\left( {\sqrt d } \right)$$ \end{document}. Given a packing of open balls with bounded radii in En and two points outside the balls at distance d from one another, the length of the shortest path connecting the two points and avoiding the balls is d + O(d/n) as d and n approaches infinity.