OP-00790 Solution claimed

A bounded number of flat transversals for fat convex sets

Fix integers $d \ge 3$, $1 \le k \le d - 2$ and $p \ge k + 2$, and a real number $\rho \ge 1$. A compact convex body $K \subset \mathbb{R}^d$ is $\rho$-fat if there are a centre $x_K$ and radii $r_K, R_K > 0$ with $$B(x_K, r_K) \subseteq K \subseteq B(x_K, R_K), \qquad R_K \le \rho\, r_K;$$ the radii may vary from body to body. A $k$-flat is a $k$-dimensional affine subspace. Jung and Pálvölgyi proved a fractional Helly theorem for $k$-flats meeting $\rho$-fat convex sets, and proved the conclusion below for families of closed balls.

Problem. Is there a constant $C = C(d, k, p, \rho)$ such that, whenever $\mathcal{F}$ is a finite family of $\rho$-fat compact convex bodies in $\mathbb{R}^d$ in which every $p$ members include $k + 2$ members met by a common $k$-flat, at most $C$ $k$-flats together meet every member of $\mathcal{F}$?

Source: Jung and Pálvölgyi (2026), Conjecture 1. Stated for finite families, d ≥ 3 and 1 ≤ k ≤ d − 2 by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: MG — Metric Geometry
Secondary: CO
Keywords: AIM Open Applied Problems, fat convex sets, geometric transversals, Helly-type theorems

References

@article{jung2026,
  author        = {Jung, Attila and P{\'a}lv{\"o}lgyi, D{\"o}m{\"o}t{\"o}r},
  title         = {k-Dimensional Transversals For Fat Convex Sets},
  journal       = {Discrete Comput. Geom.},
  year          = {2026},
  doi           = {10.1007/s00454-026-00852-8},
  eprint        = {2311.15646},
  archivePrefix = {arXiv},
}
@misc{aim-fat-convex-transversal-pq,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A bounded number of flat transversals for fat convex sets}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/596-fat-convex-transversal-pq.md},
  note         = {Entry 596 at commit 24edaad, 2026-10-15},
}

Resolution claims

Bounded flat transversals for fat convex bodies v1 GitHub ↗

Kaiwen Jin

Fix $d\geq3$, $1\leq k\leq d-2$, $p\geq k+2$, and a fatness parameter $\rho\geq1$. We prove that a finite family of $\rho$-fat compact convex bodies in $\R^d$ can be intersected by a bounded number of affine $k$-flats whenever every $p$ members contain $k+2$ members with a common $k$-flat transversal. The bound depends only on $d,k,p,\rho$; the bodies may have arbitrary sizes and …

Full resolution · not verified · fidelity not established · posted on GitHub 14 Oct 2026

Status history

When	Change	By	Reason
2026-10-18	Needs verification → Solution claimed	—	Recorded here: AIM @ 24edaad, paper/main.pdf.

Citing this problem

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

OP-00790, "A bounded number of flat transversals for fat convex sets", clopen.solutions.