Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 58, Number 3 (2014), 779-806.
Locally rich compact sets
We construct a compact metric space that has any other compact metric space as a tangent at all points, with respect to the Gromov–Hausdorff distance. 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 subset. 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 subsets is measured by the Hausdorff distance. Also the geometric properties and dimensions of such spaces and sets are studied.
Illinois J. Math., Volume 58, Number 3 (2014), 779-806.
Received: 30 June 2014
Revised: 22 December 2014
First available in Project Euclid: 9 September 2015
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Chen, Changhao; Rossi, Eino. Locally rich compact sets. Illinois J. Math. 58 (2014), no. 3, 779--806. doi:10.1215/ijm/1441790390. https://projecteuclid.org/euclid.ijm/1441790390