Geometric and analytic quasiconformality in metric measure spaces

Marked Metric Measure Spaces

A marked metric measure space (mmm-space) is a triple (X , r,μ), where (X , r) is a complete and separable metric space and μ is a probability measure on X × I for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I . It arises as a state space in the construction of Markov processes which take values in random ...

Quasiconformality, Quasisymmetry, and Removability in Loewner Spaces

where L(x,r) := sup{|f (x)−f (y)| : |x−y| ≤ r}, l(x,r) := inf {|f (x)−f (y)| : |x−y| ≥ r}, and by |x− y| we denote the distance between x and y in a metric space. We say that f is quasiconformal if there is a constant H so that H(x)≤H for every x ∈X. This infinitesimal condition is easy to state but not easy to use. For instance, it is not clear from the definition if the inverse mapping is qua...

Geometric embeddings of metric spaces

Quasi-metric Spaces with Measure

The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a Borel measure, also called an mm-space. We extend some of the mm-space concepts to the setting of a quasi-metric space with probability measure (pq-space). Our motivation comes from biological sequence comparison: we show that many common similarity measures on biologica...

Sharp Geometric and Functional Inequalities in Metric Measure Spaces with Lower Ricci Curvature Bounds

Abstract. For metric measure spaces verifying the reduced curvature-dimension condition CD∗(K,N) we prove a series of sharp functional inequalities under the additional assumption of essentially nonbranching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfy...