Domain classes covered by the mean-zero Poincare inequality

This page proves the mean-zero Poincare-Wirtinger inequality directly for bounded John domains The mean-zero Poincare inequality on bounded John domains and for bounded convex domains Poincare-Wirtinger on bounded convex domains by the direct pairwise argument, both on the same page. Bounded $C^1$ and bounded Lipschitz domains satisfy the uniform interior cone condition, and every nonempty bounded connected open set satisfying that uniform condition is a John domain (Kinnunen, Remark 5.32(1)); hence the inequality applies to those classes as well. The John-class hypothesis $1\le p<\mathrm{\infty }$ is the one used by the direct proof; no Sobolev exponent and no compactness input enter it. The bounded connected extension-domain Sobolev-Poincare form $1<p<n$ is a separate statement on this page, proved from the local mean-zero estimate and the extension-domain embedding, and it does not supersede the John-domain result.

The class inclusions and the "twisted cones" picture are Kinnunen's Remark 5.32, printed p. 141, with Laugesen's Theorem 3.29 supplying the bounded connected smooth-domain Poincare comparison. This remark records scope only: it asserts no inequality of its own, and the two direct John and convex results together with the separate extension-domain statement are proved elsewhere on this page.