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

Proximity in Triangulations and Quadrangulations