It's likely that I have a very dry sense of humour. But, if Conway was being formal he would write "To further illuminate the utility of the gluing method,..". I can't help but feel that it is written the way it is quite deliberately.

My suspicion is that c will grow with $n$ in general and that even the parallelepiped corresponding to the optimal 'Voronoi reduced basis' (i.e. the basis that yields minimum c) will need to be scaled in order for the Voronoi cell to fit inside the parralellepiped. In 2 and 3 dimensions I am sure that the parallelepiped does need to be scaled, but the optimal scaling in these dimensions, say $d$, is always less than 2 and therefore $c = \lfloor d \rfloor = 1$. I need to have a look at how Conway and Sloane prove that $c=1$ for $n \leq 3$, I haven't had a chance to do this yet.