In this paper we give upper and lower bounds as well as a heuristic estimate on the
number of vertices of the convex closure of the set $ G_n={((a,b) : a,b\in \Z,\; ab \equiv
1$ (mod $n$), $1\leq a,b\leq n-1}$. The heuristic is based on an asymptotic formula of
Renyi and Sulanke. After describing two algorithms to determine the convex closure, we
compare the numeric results with the heuristic estimate, and find that they do not
agree--there are some interesting peculiarities, for which we provide a heuristic
explanation. We then describe some numerical work on the convex closure of the graph of
random quadratic and cubic polynomials over $\Z_n$. In this case the numeric results are
in much closer agreement with the heuristic, which strongly suggests that the curve $xy=1$
(mod $n$) is ``atypical.''