  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.
