  We give sharp upper bounds on the maximal injectivity radius of finite-area
hyperbolic surfaces and use them, for each g at least 2, to identify a constant
r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces
with maximal injectivity radius at least r is compact if and only if r >
r_{g-1,2}. The main tool is a version of the "centered dual complex" that we
introduced earlier, a coarsening of the Delaunay complex of a locally finite
set. In particular, we bound the area of a compact centered dual two-cell below
given lower bounds on its side lengths.
