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PreprintCC BY-NC-ND 4.0arXiv

A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds

  • Paul Schott
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Abstract

We prove for the Euclidean space (ℝn, |.|2) the existence of subsets An ⊆ ℝn with dH(ℝn, An) arbitrarily small, such that dGH(ℝn, An) = √((n+1)/2n)dH(ℝn, An), showing that the lower bound dGH(ℝn, An) ≥ √((n+1)/2n)dH(ℝn, An) proved in arXiv:2607.18447 is tight for sets that are arbitrarily dense in the surrounding space. We also prove a similar statement for Riemannian manifolds and sufficiently dense subsets as a converse to arXiv:2309.16648.

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Cite

BibTeXRIS
BibTeX
@misc{schott2026converse,
  title = {{A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds}},
  author = {Paul Schott},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12625},
  eprint = {2609.12625},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12625},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds
AU  - Paul Schott
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12625
UR  - https://arxiv.org/abs/2609.12625
AB  - We prove for the Euclidean space (ℝ^n, |.|_2) the existence of subsets A^n ⊆ ℝ^n with d_H(ℝ^n, A^n) arbitrarily small, such that d_GH(ℝ^n, A^n) = √((n+1)/2n)d_H(ℝ^n, A^n), showing that the lower bound d_GH(ℝ^n, A^n) ≥ √((n+1)/2n)d_H(ℝ^n, A^n) proved in arXiv:2607.18447 is tight for sets that are arbitrarily dense in the surrounding space. We also prove a similar statement for Riemannian manifolds and sufficiently dense subsets as a converse to arXiv:2309.16648.
ER  -