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Towards the mathematical foundation of the minimum enclosing ball and related problems
Computational Geometry · Submitted 9 Jan 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set.
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The abstract, sentence by sentence
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. | contribution 0.85 | 0.45 | 0.10 | asserted 0.99 |
| s1 | This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. | background 1.00 | 0.08 | 0.07 | asserted 0.99 |
| s2 | The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s3 | The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. | contribution 0.76 | 0.78 | 0.32 | asserted 0.99 |
| s4 | These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries. | interpretation 0.52 | 0.10 | 0.12 | asserted 1.00 |
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The paper
Original abstract
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean space. The study of several problems that are similar or related to the minimum enclosing ball problem has received a considerable impetus from the large amount of applications of these problems in various fields of science and technology. The proposed theoretical framework is based on several enclosing (covering) and partitioning (clustering) theorems and provides among others bounds and relations between the circumradius, inradius, diameter and width of a set. These enclosing and partitioning theorems are considered as cornerstones in the field that strongly influencing developments and generalizations to other spaces and non-Euclidean geometries.
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References
How the literature uses it
4 arXiv papers cite this in 9 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 4 papers
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Therefore, it is worth mentioning basic significant results related to partitioning theorems, i.e. Helly-type theorems [ref] (see also, e.g. [ref] and the references therein).
Towards the methodology for solving the minimum enclosing ball and related problems · Promise Problems, Property Testing and Partitioning Theorems · 0.95
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The efficient computation of the SES has therefore been an area of active research, with several well-established algorithms developed over the years [ref] .
An $O(N)$ Algorithm for Solving the Smallest Enclosing Sphere Problem in the Presence of Degeneracies · Introduction · 1.00
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For a review of the topic see [ref] or [ref] .
Contractibility of the Rips complexes of Integer lattices via local domination · Local Euclidean crushing · 1.00
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