  Let $ G $ be a connected graph. If $\bar{\sigma}(v)$ denotes the arithmetic
mean of the distances from $v$ to all other vertices of $G$, then the
proximity, $\pi(G)$, of $G$ is defined as the smallest value of
$\bar{\sigma}(v)$ over all vertices $v$ of $G$. We give upper bounds for the
proximity of simple triangulations and quadrangulations of given order and
connectivity. We also construct simple triangulations and quadrangulations of
given order and connectivity that match the upper bounds asymptotically and are
likely optimal.
