  In this paper we address the question of finding the point which maximizes
the $p$-th moment of the exit time of planar Brownian motion from a given
domain. We present a geometrical method of excluding parts of the domain from
consideration which makes use of a coupling argument and the conformal
invariance of Brownian motion. In many cases the maximizing point can be
localized to a relatively small region. Several illustrative examples are
presented.
