A systolic inequality with remainder in the real projective plane (Q2053429)

In this elegant paper the authors give an elementary proof of a strengthening of Pu's systolic inequality for the real projective plane. Take any Riemannian metric \(g\) on the real projective plane. Denote the area by \(a_g\), the length of the shortest closed geodesic by \(s_g\). Pu proved \[ \frac{\frac{2}{\pi}s_g^2}{a_g}\le 1 \] with equality if and only if \(g\) is the standard round metric rescaled by a constant. Denote by \(g_0\) the standard round metric, \(f>0\) the function so that \(g=f^2g_0\), and \(v_g f\) the variance in the metric \(g\). The authors prove \[ \frac{\frac{2}{\pi}s_g^2+2\pi v_{g_0/2\pi}f}{a_g}\le 1 \] again with equality if and only if \(g\) is the standard round metric rescaled by a constant, i.e. \(f\) constant.