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?

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