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Improvement of the dimension bound for unweighted RCD spaces

  • Camillo Brena
  • Luca Gennaioli
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Abstract

We show that if (X,d,ℋn) is an RCD(K,N) space for some K∈ℝ and N∈ [1,∞), then it is also a (non-collapsed) RCD(K,n) space. In other words, unweighted finite-dimensional RCD spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when N=∞, provided that we assume in addition the n-rectifiability of the space.

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Cite

BibTeXRIS
BibTeX
@misc{brena2026improvement,
  title = {{Improvement of the dimension bound for unweighted RCD spaces}},
  author = {Camillo Brena and Luca Gennaioli},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30684},
  eprint = {2608.30684},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30684},
}
RIS
TY  - UNPB
TI  - Improvement of the dimension bound for unweighted RCD spaces
AU  - Camillo Brena
AU  - Luca Gennaioli
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30684
UR  - https://arxiv.org/abs/2608.30684
AB  - We show that if (X,d,ℋ^n) is an RCD(K,N) space for some K∈ℝ and N∈ [1,∞), then it is also a (non-collapsed) RCD(K,n) space. In other words, unweighted finite-dimensional RCD spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when N=∞, provided that we assume in addition the n-rectifiability of the space.
ER  -