## Some simple Ros type inequalities

In this short note, we prove some simple sharp geometric inequalities which are similar to Ros’s inequality.

In [Ros], Ros proved that if ${\Omega^n}$ is a smooth Riemannian manifold with boundary ${\partial \Omega}$ such that ${\partial \Omega}$ has positive normalized mean curvature ${\sigma}$ and ${\Omega}$ has non-negative Ricci curvature, then

$\displaystyle n \mathrm{Vol}(\Omega)\le \int _{\partial \Omega}\frac{1}{\sigma}d\mu.$

The equality holds if and only if $\Omega$ is an Euclidean ball.

 Theorem 1 Suppose ${\gamma}$ is a smooth simple closed curve embedded in ${M=\mathbb{R}^2}$, ${\mathbb{S}^2}$ or ${\mathbb{H}^2}$, such that its geodesic curvature is positive. Let ${A}$ be the area of the region bounded by ${\gamma}$, ${L=\mathrm{Length}(\gamma)}$, ${K}$ be the curvature of ${M}$, then $\displaystyle \int _{\gamma}\frac{1}{k}ds \ge \frac{L^2}{2\pi-K A}\ge \frac{4\pi A-K A^2}{2\pi -K A}.$ The equality holds if and only if ${\gamma}$ is a geodesic circle.

Proof: The first equality follows by combining the Cauchy-Schwarz inequality

$\displaystyle L^2 =\left(\int_{\gamma}1 ds\right)^2\le \int _{\gamma}kds \int \frac{1}{k}ds$

and the Gauss-Bonnet theorem

$\displaystyle 2 \pi=2\pi \chi (\Omega) = \int_\Omega K dS + \int_\gamma k ds = KA+ \int_\gamma kds,$

where ${\Omega}$ is the region bounded by ${\gamma}$. The second inequality is the isoperimetric inequality

$\displaystyle L^2 \ge 4\pi A-KA^2.$

It is clear from the Cauchy-Schwarz inequality (or the isoperimetric inequality) that the equality holds if and only if ${k}$ is constant, and so ${\gamma}$ is a geodesic circle. $\Box$

Question: What’s the analogous inequality in higher dimension?