Sharp barrier-parameter growth for hyperbolic balls
Let $\mathbb{H}^2$ be the hyperbolic plane of curvature $-1$, fix $o \in \mathbb{H}^2$ and let $D_R = \{x \in \mathbb{H}^2 : d(o, x) < R\}$. Covariant derivatives are those of the Levi-Civita connection. A $\theta$-barrier on $D_R$ is a function $F \in C^3(D_R)$ with positive definite Hessian $H_x = (\nabla^2 F)_x$ and with closed epigraph in $\mathbb{H}^2 \times \mathbb{R}$, such that for every $x \in D_R$ and all $u, v, w \in T_x \mathbb{H}^2$ $$|(\nabla^3 F)_x(u, v, w)| \le 2\sqrt{H_x(u,u)\, H_x(v,v)\, H_x(w,w)}, \qquad |dF_x(v)|^2 \le \theta\, H_x(v,v).$$ The first inequality is required for all triples, not only for $u = v = w$. Let $$\theta_*(R) = \inf\{\theta \ge 0 : \text{a } \theta\text{-barrier on } D_R \text{ exists}\};$$ the barrier may depend on $R$ and need not be rotationally symmetric. These are the barriers of the interior-point methods on Hadamard manifolds of Hirai, Nieuwboer and Walter, whose construction gives $\theta_*(R) = O(R^2)$. Criscitiello, Nieuwboer and Walter proved that $\theta_*(R) \ge \sqrt{R}/128 - 1/4$, even when the first inequality is required only for $u = v = w$. In Euclidean space, by contrast, balls of every radius admit barriers whose parameter does not depend on the radius.
Problem. Determine the asymptotic order of $\theta_*(R)$ as $R \to \infty$, up to multiplicative constants independent of $R$. In particular, is $\theta_*(R) = O(\sqrt{R})$?
Source: Criscitiello, Nieuwboer and Walter (2026), after Corollary 1.4, who ask whether their lower bound Ω(√R) for balls is tight; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, Hadamard manifolds, interior-point methods, self-concordant barriers
References
@misc{criscitiello2026,
author = {Criscitiello, Christopher and Nieuwboer, Harold and Walter, Michael},
title = {Negative curvature obstructs the existence of good barriers for interior-point methods},
year = {2026},
eprint = {2604.07319},
archivePrefix = {arXiv},
}
@article{hirai2026,
author = {Hirai, Hiroshi and Nieuwboer, Harold and Walter, Michael},
title = {Interior-point methods on manifolds: theory and applications},
journal = {Found. Comput. Math.},
year = {2026},
doi = {10.1007/s10208-026-09756-8},
eprint = {2303.04771},
archivePrefix = {arXiv},
}
@misc{aim-hyperbolic-ball-barrier-parameter,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp barrier-parameter growth for hyperbolic balls}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/540-hyperbolic-ball-barrier-parameter.md},
note = {Entry 540 at commit a153f46, 2026-10-13},
}
Resolution claims
For geodesic balls of radius \(R\) in the hyperbolic plane of curvature \(-1\), we prove that the least parameter of a self-concordant barrier is of order \(R^2\). Self-concordance is required for arbitrary triples of tangent vectors. The lower bound holds for every barrier, without a symmetry assumption: at a maximum on a geodesic circle, curvature forces a fixed positive radial …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Quadratic barrier parameters for hyperbolic balls v1 recorded from AIM @ 24edaad, paper/hyperbolic_ball_barriers.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00354, "Sharp barrier-parameter growth for hyperbolic balls", clopen.solutions.