On Local Poincare via Transportation

It is shown that curvature-dimension bounds CD(N, k) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L^1- Poincare-inequality under some symmetry assumption on the choice of transport rays in the cut locus of (X,d). This condition is satisfied if (X,d) has m-almost surely no branching points.