In the 1950s, Carl Loewner proved an inequality relating the shortest closed geodesics on a
2-torus to its area. Many generalisations have been developed since, by Gromov and others. We
show that the shortest closed geodesic on an area-minimizing surface S for a generic metric on
CP^2 is controlled by the total volume, even though the area of S is not. We exploit the
Croke--Rotman inequality, Gromov's systolic inequalities, the Kronheimer--Mrowka proof of the
Thom conjecture, and White's regularity results for area minimizers.
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