# Area of convex disks

## TL;DR

This paper proves a lower bound on the area of metric balls in two-dimensional Riemannian manifolds for radii less than half the convexity radius, confirming a long-standing conjecture and deriving bounds on Laplacian eigenvalues.

## Contribution

It establishes the conjectured area inequality for metric balls and derives an eigenvalue upper bound, advancing understanding in Riemannian geometry.

## Key findings

- Proved that the area of metric balls is at least (8/π) R^2 for R less than half the convexity radius.
- Derived an upper bound on the first nonzero Neumann eigenvalue of the Laplacian based on the radius.
- Confirmed long-standing conjectures relating geometry and spectral properties of Riemannian manifolds.

## 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 $Area(B(p,R)) \geq \frac{8}{\pi}R^2$. 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.