Geometric implications of the Poincaré inequality. (English) Zbl 1146.46019
Summary: The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak \((1, p)\)-Poincaré inequality with \(p\) sufficiently small, then annuli are almost quasiconvex. We also obtain estimates for the Hausdorff \(s\)-content and the diameter of the spheres.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 31C15 | Potentials and capacities on other spaces |
