Geometric implications of the poincare inequality

The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak (1, p)-Poincare 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. Mathematics Subject Classification (2000). Primary 46E35; Secondary 31C15.