Area of convex disks

Abstract

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 $Area(B(p,R))\ge \frac{8}{\pi }{R}^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound ${\mu }_2(B(p,R))\le 2(\frac{\pi }{2R}{)}^2$ 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.

4 pages