135 research outputs found
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
there is a doubling measure having full measure on a set of
packing dimension at most
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