Packing $d$-dimensional balls into a $d + 1$-dimensional container

Alt, Helmut (author), Cabello, Sergio (author), Cheong, Otfried (author), Park, Ji-won (author), Seiferth, Nadja (author)

In this article, we consider the problems of finding in $d + 1$ dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of $d$-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for $n$ such balls, a container of volume $O(n^{d−1 \over d})$ is always sufficient and sometimes necessary. As a byproduct, this implies that for $d \ge 2$ there is no finite size $(d + 1)$-dimensional convex body into which all $d$-dimensional unit-radius balls can be packed simultaneously.

• Language: English
• Keywords: packings, translations, unit disks, unit balls, volume
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: 13 str.
• Numbering: Vol. 132, art. 102219
• PID: 20.500.12556/RUL-176508
• UDC: 004.92:514
• ISSN on article: 0925-7721
• DOI: 10.1016/j.comgeo.2025.102219
• COBISS.SI-ID: 259634179
• Publication date in RUL: 02.12.2025

Record is a part of a journal

• Title: Computational geometry
• Shortened title: Comput. geom.
• Publisher: Elsevier
• ISSN: 0925-7721
• COBISS.SI-ID: 14903301

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.