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On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (2013)

Annales scientifiques de l'École Normale Supérieure

In this article we study the Ahlfors regular conformal gauge of a compact metric space ( X , d ) , and its conformal dimension dim A R ( X , d ) . Using a sequence of finite coverings of  ( X , d ) , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute dim A R ( X , d ) using the critical exponent Q N associated to the combinatorial modulus.

On the Regularity of Alexandrov Surfaces with Curvature Bounded Below

Luigi Ambrosio, Jérôme Bertrand (2016)

Analysis and Geometry in Metric Spaces

In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.

Ramsey partitions and proximity data structures

Manor Mendel, Assaf Naor (2007)

Journal of the European Mathematical Society

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion.We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem,...

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