A Poincar\'{e}-type inequality and a related eigenvalue problem

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses Fenchel’s theorem about the total curvature of closed space curves in an essential way. Next we consider the generalization of this result to higher dimensional closed Riemannian manifold and reduce it to an eigenvalue problem. Finally, we point out that even though such Poincar'{e}-type inequality still holds, the best constant ${\lambda }_1(f)$ might be different from the first eigenvalue ${\lambda }_1$ by constructing explicit examples on the standard spheres and flat tori.