On the geometry of the smallest circle enclosing a finite set of points

Drager, Lance D.; Lee, Jeffrey M.; Martin, Clyde F.

doi:10.1016/j.jfranklin.2007.01.003

Cited by 11 publications

(6 citation statements)

References 4 publications

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“…This is called the smallest enclosing circle problem, which was introduced by Sylvester [18]. For more information, see [5,13,20]. It is an open problem that d is a g-metric for any n ≥ 3.…”

Section: Theory Of a G-metricmentioning

confidence: 99%

Structure for $g$-Metric Spaces and Related Fixed Point Theorems

Choi,

Kim,

Yang

2018

Preprint

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In this paper, we propose a generalized notion of a distance function, called a g-metric.The g-metric with degree n is a distance of n + 1 points, generalizing the ordinary distance between two points and G-metric between three points. Indeed, it is shown that the g-metric with degree 1 (resp. degree 2) is equivalent to the ordinary metric (resp. the G-metric). Fundamental properties and several examples for the g-metric are also given. Moreover, topological properties on the g-metric space including the convergence of sequences and the continuity of mappings on the g-metric space are studied. Finally, we generalize some well-known fixed point theorems including Banach contraction mapping principle and Ćirić fixed point theorem in the g-metric space.

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Section: Theory Of a G-metricmentioning

confidence: 99%

Structure for $g$-Metric Spaces and Related Fixed Point Theorems

Choi,

Kim,

Yang

2018

Preprint

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“…Over a century later, the smallest enclosing circle problem remains very active due to its important applications to clustering, nearest neighbor search, data classification, facility location, collision detection, computer graphics, and military operations. The problem has been widely treated in the literature from both theoretical and numerical standpoints; see [1,4,6,7,9,21,23,25,27,28,31] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

The Log-Exponential Smoothing Technique and Nesterov’s Accelerated Gradient Method for Generalized Sylvester Problems

An¹,

Giles

Nam

et al. 2015

J Optim Theory Appl

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The Sylvester smallest enclosing circle problem involves finding the smallest circle that encloses a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov's accelerated gradient method, we present an effective numerical algorithm for solving these problems.

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“…The Sylvester problem and its version in higher dimensions are now called under different names such as: the smallest enclosing ball problem, the minimum ball problem, or the bomb problem. The readers are referred to [2][3][4][5][6][7] and the references therein for recent study on the problem and its generalizations.…”

Section: Introductionmentioning

confidence: 99%

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

Nam¹,

Hoàng²,

An³

2013

J Optim Theory Appl

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The classical Apollonius' problem is to construct circles that are tangent to three given circles in a plane. This problem was posed by Apollonius of Perga in his work "Tangencies". The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls in R n , find the smallest Euclidean ball that encloses all of the balls in the first collection and intersects all of the balls in the second collection. We also study a generalized version of the Fermat-Torricelli problem stated as follows: given two finite collections composed of three Euclidean balls in R n , find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.

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On the geometry of the smallest circle enclosing a finite set of points

Cited by 11 publications

References 4 publications

Structure for $g$-Metric Spaces and Related Fixed Point Theorems

Structure for $g$-Metric Spaces and Related Fixed Point Theorems

The Log-Exponential Smoothing Technique and Nesterov’s Accelerated Gradient Method for Generalized Sylvester Problems

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

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