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An optimal base station (BS) location depends on the traffic (user) distribution, propagation pathloss and many system parameters, which renders its analytical study difficult so that numerical algorithms are widely used instead. In this paper, the problem is studied analytically. First, it is formulated as a convex optimization problem to minimize the total BS transmit power subject to quality-of-service (QoS) constraints, which also account for fairness among users. Due to its convex nature, Karush-Kuhn-Tucker (KKT) conditions are used to characterize a globally-optimum location as a convex combination of user locations, where convex weights depend on user parameters, pathloss exponent and overall geometry of the problem. Based on this characterization, a number of closed-form solutions are obtained. In particular, the optimum BS location is the mean of user locations in the case of free-space propagation and identical user parameters. If the user set is symmetric (as defined in the paper), the optimal BS location is independent of pathloss exponent, which is not the case in general. The analytical results show the impact of propagation conditions as well as system and user parameters on optimal BS location and can be used to develop design guidelines.