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arXiv · 1406.7643

Locally rich compact sets

Changhao Chen·Eino Rossi
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Abstract

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a dense sub-set. Here the "almost all compact sets" means that the tangent collection contains a contracted image of any compact set of the cube and that the contraction ratios are uniformly bounded. In the Euclidean space, the distance of sub-sets is measured by the Hausdorff distance. Also the geometric properties and dimensions of such spaces and sets are studied.

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Changhao Chen, Eino Rossi. 2014-06-30. Locally rich compact sets. https://doi.org/10.1215/ijm/1441790390

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