Poincare inequality on an annulus (${\int }_{A_R}|v-v_{A_R}{|}^2\le C{R}^2{\int }_{A_R}|\nabla v{|}^2$)

For $n\ge 2$, a round annulus $A_R=B_{2R}\setminus B_R$ is connected and obeys ${\int }_{A_R}|v-v_{A_R}{|}^2\le C(n)R^2{\int }_{A_R}|\nabla v{|}^2$. Scaling reduces the claim to a fixed annulus. In dimension one the annulus has two components, so the inequality fails with a single average; one needs a separate average on each component.