## Illinois Journal of Mathematics

### Locally rich compact sets

#### Abstract

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.

#### Article information

Source
Illinois J. Math., Volume 58, Number 3 (2014), 779-806.

Dates
Revised: 22 December 2014
First available in Project Euclid: 9 September 2015

https://projecteuclid.org/euclid.ijm/1441790390

Digital Object Identifier
doi:10.1215/ijm/1441790390

Mathematical Reviews number (MathSciNet)
MR3395963

Zentralblatt MATH identifier
1330.53054

Subjects
Secondary: 37F40: Geometric limits

#### Citation

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

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