In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile is defined by $h(\psi)=\inf ( |\partial\Omega|:\Omega\subset M, |\Omega|=\psi|M| )$

Theorem 2: Let $M$ be a smooth compact Riemannian surface. Then the isoperimetric profile satisfies $$h(\psi)=\sqrt{4\pi|M|\psi} - \frac{|M|^{3/2}\sup_MK}{4\sqrt{\pi}}\psi^{3/2} + O(\psi^2)$$ as $\psi\to 0$.

In the proof of above theorem, they say that "The upper bound follows since $|\partial B_r(p)|\geq h(|B_r(p)|/|M|).$ But I don't know how this inequality implies the upper bound for the equality in Theorem 2.

Does anyone read this paper? Do you know how to achieve this. Thanks.

In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile is defined by $h(\psi)=\inf ( |\partial\Omega|:\Omega\subset M, |\Omega|=\psi|M| )$

Theorem 2: Let $M$ be a smooth compact Riemannian surface. Then the isoperimetric profile satisfies $$h(\psi)=\sqrt{4\pi|M|\psi} - \frac{|M|^{3/2}\sup_MK}{4\sqrt{\pi}}\psi^{3/2} + O(\psi^2)$$ as $\psi\to 0$.

In the proof of above theorem, they say that "The upper bound follows since $|\partial B_r(p)|\geq h(|B_r(p)|/|M|).$ But I don't know how this inequality implies the upper bound for the equality in Theorem 2.

Does anyone read this paper? Do you know how to achieve this? Thanks.