Covering Balls by Translated Slabs

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$ satisfy $d>2$. Equip ${\mathbb{R}}^d$ with its Euclidean inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|_2$, and define \[ S^{d-1}=\{n\in{\mathbb{R}}^d:\|n\|_2=1\},\qquad \overline B_d=\{x\in{\mathbb{R}}^d:\|x\|_2\leq1\}. \] For $n\in S^{d-1}$, $\varepsilon>0$, and $a\in{\mathbb{R}}$, define the closed slab of width $\varepsilon$, normal direction $n$, and offset $a$ by \[ P(n,\varepsilon,a)=\{x\in{\mathbb{R}}^d:a\leq\langle x,n\rangle\leq a+\varepsilon\}. \] Changing $a$ allows an arbitrary translate of the slab while preserving its normal direction and width.

The finite covering question described in the supplied record is whether, for every integer $d>2$, there exists a finite constant $C(d)>0$ such that, for every positive integer $m$, every choice of directions $n_1,\ldots,n_m\in S^{d-1}$, and every choice of widths $\varepsilon_1,\ldots,\varepsilon_m>0$, \[ \sum_{i=1}^{m}\varepsilon_i>C(d) \quad\Longrightarrow\quad \exists(a_1,\ldots,a_m)\in{\mathbb{R}}^m\quad \overline B_d\subseteq\bigcup_{i=1}^{m}P(n_i,\varepsilon_i,a_i). \] Thus $C(d)$ must depend only on the dimension; the offsets may depend on the entire given family. The displayed slab inequalities and the strict inequality on the total width are retained exactly.

Formulation qualifications. The source statements do not specify an index set, but the supplied literature assessment explicitly calls the target a finite slab-translate covering question; the formulation above follows that qualification. The source says "unit ball" without specifying whether its boundary is included. The display above explicitly uses the closed unit ball. If the intended convention is the open unit ball, replace $\overline B_d$ by $B_d^{\circ}=\{x\in{\mathbb{R}}^d:\|x\|_2<1\}$; the intended boundary convention requires confirmation.

Short English statement

For each integer dimension $d>2$, is there a threshold depending only on $d$ such that every finite family of slabs with prescribed normal directions and positive widths whose sum exceeds that threshold can be translated to cover the unit ball? The supplied curation labels this question open. The source does not specify whether the ball includes its boundary; the displayed formulation uses the closed ball.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003470$, title "Covering Balls by Translated Slabs," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2017 and names no individual proposer.
• Supplied source citation: Discrete Geometry (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/19, pp. 1135--1205. https://doi.org/10.4171/owr/2017/19. This is also the supplied literature link labelled "original OWR report." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Kupavskii and Pach on plank covering." https://arxiv.org/abs/1511.08111.
• Literature assessment dated 2026-08-21: the supplied curation labels the finite slab-translate covering question unresolved and attributes to Kupavskii and Pach a weaker infinite covering criterion and a resolution of a related polynomial approximation problem. This is supplied dated curation, not an independently verified assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. These are supplied editorial labels and assertions, not independent mathematical verification of this draft.