  We prove an inequality of Bonnesen type for the real projective plane,
generalizing Pu's systolic inequality for positively-curved metrics. The
remainder term in the inequality, analogous to that in Bonnesen's inequality,
is a function of R-r (suitably normalized), where R and r are respectively the
circumradius and the inradius of the Weyl-Lewy Euclidean embedding of the
orientable double cover. We exploit John ellipsoids of a convex body and
Pogorelov's ridigity theorem.
