Pu's inequality for the real projective plane
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In differential geometry, in systolic geometry, Pu's inequality applies to general Riemannian metrics, on the space obtained by identifying antipodal points on the 2-sphere.
A student of Charles Loewner's, P.M. Pu proved in a 1950 thesis (published in 1952) that every metric on the real projective plane 
satisfies the optimal inequality
where sys is the systole. The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature.
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[edit] Reformulation
Alternatively, every metric on the sphere S2 invariant under the antipodal map admits a pair of opposite points 
at Riemannian distance d = d(p,q) satisfying 
A more detailed explanation of this viewpoint may be found at the page Introduction to systolic geometry.
[edit] Pu's inequality and the filling area conjecture
An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length 2π by a 2-dimensional disk with the strongly isometric property, the round hemisphere has the least area.
To explain this formulation, we start with the observation that the equatorial circle of the unit 2-sphere 
is a Riemannian circle S1 of length 2π. More precisely, the Riemannian distance function of S1 is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is only 2, whereas in the Riemannian circle it is π.
We consider all fillings of S1 by a 2-dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian metric of a circle of length 2π. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.
In 1983 Gromov conjectured that the round hemisphere gives the "best" way of filling the circle even when the filling surface is allowed to have positive genus.
[edit] Pu's inequality and isoperimetric inequality
Pu's inequality bears a curious resemblance to the classical isoperimetric inequality
for Jordan curves in the plane, where L is the length of the curve while A is the area of the region it bounds. Namely, in both cases a 2-dimensional quantity (area) is bounded by (the square of) a 1-dimensional quantity (length). However, the inequality goes in the opposite direction. Thus, Pu's inequality can be thought of as an "opposite" isoperimetric inequality.
