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Fractal dimensions in the Gromov–Hausdorff space

Volume 71 / 2023

Yoshito Ishiki Bulletin Polish Acad. Sci. Math. 71 (2023), 147-168 MSC: Primary 53C23; Secondary 51F99 DOI: 10.4064/ba221212-16-11 Published online: 11 December 2023


We first show that for any four non-negative real numbers, there exists a Cantor ultrametric space whose Hausdorff dimension, packing dimension, upper box dimension, and Assouad dimension are equal to the given four numbers, respectively. Next, using a direct sum of metric spaces, we construct topological embeddings of an arbitrary compact metrizable space into the two subsets of the Gromov–Hausdorff space: the set of all compact metric spaces possessing prescribed topological dimension and the aforementioned four dimensions, and the set of all compact ultrametric spaces.


  • Yoshito IshikiPhotonics Control Technology Team
    RIKEN Center for Advanced Photonics
    Wako, Saitama 351-0198, Japan

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