227 research outputs found
Locally rich compact sets
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.Comment: 29 pages, 3 figures. Final versio
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