In this short note, we prove some simple sharp geometric inequalities which are similar to Ros’s inequality.
In [Ros], Ros proved that if is a smooth Riemannian manifold with boundary such that has positive normalized mean curvature and has non-negative Ricci curvature, then
The equality holds if and only if is an Euclidean ball.
Theorem 1 Suppose is a smooth simple closed curve embedded in , or , such that its geodesic curvature is positive. Let be the area of the region bounded by , , be the curvature of , then
The equality holds if and only if is a geodesic circle.
Proof: The first equality follows by combining the Cauchy-Schwarz inequality
and the Gauss-Bonnet theorem
where is the region bounded by . The second inequality is the isoperimetric inequality
It is clear from the Cauchy-Schwarz inequality (or the isoperimetric inequality) that the equality holds if and only if is constant, and so is a geodesic circle.
Question: What’s the analogous inequality in higher dimension?