Illinois Journal of Mathematics

Locally rich compact sets

Changhao Chen and Eino Rossi

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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

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A80: Fractals [See also 37Fxx]
Secondary: 37F40: Geometric limits


Chen, Changhao; Rossi, Eino. Locally rich compact sets. Illinois J. Math. 58 (2014), no. 3, 779--806. doi:10.1215/ijm/1441790390.

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