OP-00672 Solution claimed

The cost of continuous reconstruction on Banach unit balls

Let $X$ and $Y$ be real Banach spaces, let $B_X$ be the closed unit ball of $X$ with its norm topology, and let $S : X \to Y$ be a bounded linear operator. For $n \ge 1$ let $$e_n(S) = \inf_{N, \phi} \sup_{x \in B_X} \|Sx - \phi(N(x))\|_Y,$$ the infimum over continuous maps $N : B_X \to \mathbb{R}^n$ and arbitrary maps $\phi : \mathbb{R}^n \to Y$, and let the manifold width $\delta_n(S)$ be the same infimum with $\phi$ also required to be continuous on all of $\mathbb{R}^n$. The measurement map $N$ is a single non-adaptive map, both infima optimise it together with the reconstruction, and no Lipschitz bound is imposed on either map; clearly $e_n(S) \le \delta_n(S)$. Krieg, Novak and Ullrich proved that $\delta_n(S) \le 2 e_n(S)$ when $B_X$ is replaced by a compact metric space $F$ and $S : F \to Y$ is any continuous map into a normed space, and that $e_n(S) = \delta_n(S)$ when $X$ and $Y$ are Hilbert spaces.

Problem. Is there a universal constant $C < \infty$ such that $\delta_n(S) \le C e_n(S)$ for all real Banach spaces $X$ and $Y$, every bounded linear $S : X \to Y$ and every $n \ge 1$? In particular, does $C = 2$ suffice?

Source: Stated by AIM — Open Applied Problems, after Krieg and Ullrich (2026), who, after their Proposition 9.5 (from Krieg, Novak and Ullrich, Lemma 9), do not know whether compactness and the factor 2 are needed in general.

Imported from github.com/MColbrook/AIM.

Classification

Primary: NA — Numerical Analysis
Secondary: FA
Keywords: AIM Open Applied Problems, continuous measurements, information-based complexity, manifold widths

References

@article{krieg2026acta,
  author        = {Krieg, David and Ullrich, Mario},
  title         = {Approximation of functions: optimal sampling and complexity},
  journal       = {Acta Numer.},
  volume        = {35},
  year          = {2026},
  pages         = {273--457},
  doi           = {10.1017/S0962492925100287},
  eprint        = {2602.02066},
  archivePrefix = {arXiv},
}
@misc{krieg2026continuous,
  author        = {Krieg, David and Novak, Erich and Ullrich, Mario},
  title         = {How many continuous measurements are needed to learn a vector?},
  year          = {2026},
  eprint        = {2412.06468},
  archivePrefix = {arXiv},
  note          = {Version 2, 22 September 2026},
}
@misc{aim-continuous-decoder-banach-balls,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The cost of continuous reconstruction on Banach unit balls}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/570-continuous-decoder-banach-balls.md},
  note         = {Entry 570 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Continuous reconstruction on Banach unit balls v1 GitHub ↗

Siavash Sadeghi

One fixed inclusion from $\ell_1([0,1])$ to $\ell_2([0,1])$ disproves every universal continuity-cost constant over arbitrary real Banach spaces. The construction uses nonseparable spaces and does not settle a separable variant.

Counterexample · not verified · fidelity not established · posted on GitHub 4 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, siavash-sadeghi-570/aim570_report.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00672, "The cost of continuous reconstruction on Banach unit balls", clopen.solutions.