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Optimal-diameter partitions into regions of the prescribed measure

  • Grigory Voinov
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Abstract

In this paper we find a sufficient condition on an Ahlfors--David regular metric measure space under which it admits a partition into parts of prescribed measures and optimal (up to a constant) diameters. The proof uses the construction of dyadic cubes. The process is algorithmic: the pieces are cut out one by one via a filling procedure on the tree of dyadic cubes. We introduce the notion of spaces which admit a connected dyadic cube decomposition and prove that they admit a partition of the kind described above. We then develop several techniques to obtain such spaces and show some natural examples of this type.

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Cite

BibTeXRIS
BibTeX
@misc{voinov2026optimaldiameter,
  title = {{Optimal-diameter partitions into regions of the prescribed measure}},
  author = {Grigory Voinov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23867},
  eprint = {2609.23867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23867},
}
RIS
TY  - UNPB
TI  - Optimal-diameter partitions into regions of the prescribed measure
AU  - Grigory Voinov
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23867
UR  - https://arxiv.org/abs/2609.23867
AB  - In this paper we find a sufficient condition on an Ahlfors--David regular metric measure space under which it admits a partition into parts of prescribed measures and optimal (up to a constant) diameters. The proof uses the construction of dyadic cubes. The process is algorithmic: the pieces are cut out one by one via a filling procedure on the tree of dyadic cubes. We introduce the notion of spaces which admit a connected dyadic cube decomposition and prove that they admit a partition of the kind described above. We then develop several techniques to obtain such spaces and show some natural examples of this type.
ER  -