  This paper considers metric balls $B(p,R)$ in two dimensional Riemannian
manifolds when $R$ is less than half the convexity radius. We prove that
$Area(B(p,R)) \geq \frac{8}{\pi}R^2$. This inequality has long been conjectured
for $R$ less than half the injectivity radius. This result also yields the
upper bound $\mu_2(B(p,R)) \leq 2(\frac{\pi}{2 R})^2$ on the first nonzero
Neumann eigenvalue $\mu_2$ of the Laplacian in terms only of the radius. This
has also been conjectured for $R$ up to half the injectivity radius.
