Coxeter groups are almost convex (Q1814109)

According to \textit{J. W. Cannon} [Geom. Dedicata 22, 197-210 (1987; Zbl 0607.20020)] a group \(G\) is called almost convex with respect to a generating set \(S=S^{-1}\) if the Cayley graph \(\Gamma(G,C)\) is almost convex, that is, if there is \(N(k)\) for some number \(k\) such that elements \(x\), \(y\) of distance \(k\) from 1 can be connected by a path of length \(N(k)\) using elements of length at most \(k\) only. The authors show that this condition is satisfied for Coxeter groups with respect to the usual set of involutory generators.