The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

Mordukhovich, Boris S.; Nam, Nguyen Mau; Villalobos, C.I

doi:10.1007/s11590-012-0483-7

Cited by 18 publications

(11 citation statements)

References 12 publications

(10 reference statements)

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“…Given a normed vector space V and a proper finite subset S ⊂ V , a minimal enclosing ball for S is a closed ball B ⊂ V which contains S and such that any other ball containing S have a bigger radius. Using classic convex optimization results, one can show the existence and uniqueness of a minimal enclosing ball in the case of rotund reflexive normed vector spaces [14]. One may wonder whether the center c S and the radius r S of the minimal enclosing ball of a proper finite subset S vary continuously with S. This question is naturally posed using the metric exponential Exp M (V ).…”

Section: Interlude: Minimal Enclosing Ballsmentioning

confidence: 99%

On the topologies of the exponential

Cepek,

Lejay

2021

Preprint

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Factorization algebras have been defined using three different topologies on the Ran space. We study these three different topologies on the exponential, which is the union of the Ran space and the empty configuration, and show that an exponential property is satisfied in each case. As a consequence we describe the weak homotopy type of the exponential Exp(X) for each topology, in the case where X is not connected.We also study these exponentials as stratified spaces and show that the metric exponential is conically stratified for a general class of spaces. As a corollary, we obtain that locally constant factorization algebras defined by Beilinson-Drinfeld are equivalent to locally constant factorization algebras defined by Lurie. contents 1 Exponentials 3 1.1 Exponential functors 3 1.2 The exponential functor 6 2 Topologies on the exponential 9 2.1 The topological exponential 9 2.1.1 First steps towards exponentiability 9 2.1.2 The topological exponential is not an exponential 10 2.1.3 The topological exponential is almost an exponential 11 2.1.4 The topological exponential is a restricted exponential 11 2.2 The metric exponential 12 2.3 The minimal exponential 14 2.4 Weak homotopy type of the exponentials 15 3 Interlude: minimal enclosing balls 17 4 Stratification of the exponentials 19 4.1 The exponentials as stratified spaces 19 4.2 Cones and joins 20 4.3 Conical stratification 21 This work was supported by IBS-R003-D1.

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Section: Interlude: Minimal Enclosing Ballsmentioning

confidence: 99%

On the topologies of the exponential

Cepek,

Lejay

2021

Preprint

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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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“…We shall refer to this problem as the largest enclosed ball problem (see Figure 4). For previous work on this problem see [18,19] and the references therein. The smallest intersecting and enclosed ball problem.…”

Section: Equivalent Geometric Problemsmentioning

confidence: 99%

A dual Simplex-type algorithm for the smallest enclosing ball of balls and related problems

Cavaleiro,

Alizadeh

2018

Preprint

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The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

Cited by 18 publications

References 12 publications

On the topologies of the exponential

On the topologies of the exponential

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

A dual Simplex-type algorithm for the smallest enclosing ball of balls and related problems

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