Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincar\'e inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincar\'e inequality is implied by the weak displacement convexity of the functional.Comment: 25 pages, 1 figur

Approximation by uniform domains in doubling quasiconvex metric spaces

We show that any bounded domain in a doubling quasiconvex metric space can be approximated from inside and outside by uniform domains.Comment: 7 page

Existence of doubling measures via generalised nested cubes

Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each $\epsilon>0$ there is a doubling measure having full measure on a set of packing dimension at most $\epsilon$