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1
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Products of Snowflaked Euclidean Lines Are Not Minimal for Looking Down

100%
Joseph M., Rajala T.
Analysis and Geometry in Metric Spaces
|
2017
|
tom 5
|
nr 1
78-97
EN
We show that products of snowflaked Euclidean lines are not minimal for looking down. This question was raised in Fractured fractals and broken dreams, Problem 11.17, by David and Semmes. The proof uses arguments developed by Le Donne, Li and Rajala to prove that the Heisenberg group is not minimal for looking down. By a method of shortcuts, we define a new distance d such that the product of snowflaked Euclidean lines looks down on (RN , d), but not vice versa.
2
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Tangent Lines and Lipschitz Differentiability Spaces

100%
Cavalletti F., Rajala T.
Analysis and Geometry in Metric Spaces
|
2016
|
tom 4
|
nr 1
EN
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.
3
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Thin and fat sets for doubling measures in metric spaces

81%
Ojala T., Rajala T., Suomala V.
Studia Mathematica
|
2012
|
tom 208
|
nr 3
195-211
EN
We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.
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