Numerical Geometry-Numbers for Shapes

Cleary, Joan; Morris, Sidney A.; Yost, David

doi:10.2307/2323675

Cited by 18 publications

(29 citation statements)

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“…Note that Elton did not publish his result, but references to his work [4,14] mention that he proved his statement even for continuous, nonnegative and symmetric functions f (in place of the metric d) over compact connected Hausdorff topological spaces. In any case, his results are now included in the following.…”

Section: Rendezvous Numbers For Metric Spacesmentioning

confidence: 99%

“…Since the first result of Gross [12], rendezvous numbers have been attracting much attention and been generalized in many directions: considering weak rendezvous numbers (Thomassen [20]), replacing the metric by some continuous symmetric function (Stadje [18]) or considering instead of the finite average in (1.1) the mean value with respect to some probability measure µ (Elton, Cleary, Morris, Yost [4]). In such abstract investigations various minimax principles play important role.…”

Section: Introductionmentioning

confidence: 99%

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Rendezvous numbers of metric spaces – a potential theoretic approach

Farkas

Révész

2006

Arch. Math.

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Abstract. The present work draws on the understanding how notions of general potential theory -as set up, e.g., by Fuglede -explain existence and some basic results on the "magical" rendezvous numbers. We aim at a fairly general description of rendezvous numbers in a metric space by using systematically the potential theoretic approach.In particular, we generalize and explain results on invariant measures, hypermetric spaces and maximal energy measures, when showing how more general proofs can be found to them. IntroductionIt was proved by O. Gross that for a compact, connected metric space (X, d) there exists a unique number r = r(X) such that for every finite point system x 1 , . . . , x n ∈ X, n ∈ N one always finds an x ∈ X with Our aim is to put the investigations on the existence and uniqueness of rendezvous numbers in the framework of abstract potential theory, which has been around since the 60s, but apparently has not gained its due recognition in this field. In this paper we continue [6] First we spend some words on technicalities and recall the appropriate setting of potential theory in locally compact spaces. For convenience we add +∞ to the set of real numbers, i.e.,2000 Mathematics Subject Classification. Primary: 31C15. Secondary: 28A12, 54D45. Key words and phrases. Locally compact Hausdorff topological spaces, potential theoretic kernel function in the sense of Fuglede, potential of a measure, energy integral, energy and capacity of a set, Chebyshev constant, (weak) rendezvous number, average distance, minimax theorem, invariant measure, positive definite kernel, maximum principle, Frostman's equilibrium theorem.

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Section: Rendezvous Numbers For Metric Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rendezvous numbers of metric spaces – a potential theoretic approach

Farkas

Révész

2006

Arch. Math.

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“…Now we recall a result of Gross which is an easy consequence of Theorem A or one of its ancestors due to Glicksberg [7] or to Peck-Dulmage [16] (compare [8] and also [4], [14], [19]). …”

Section: Convex Metric Spacesmentioning

confidence: 99%

“…We use this theorem to prove the following generalization of a result of Esther and George Szekeres ( [4], Theorem 5) and of Yang Lu and Zhang Jingzong [21]: (9) ψ(0, α) + ψ(α, 0) ≤ α, 0 < α ≤ δ(X).…”

Section: Convex Metric Spacesmentioning

confidence: 99%

Minimax theorems with applications to convex metric spaces

Kindler¹

1995

Colloq. Math.

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“…Recently the work of Gross [4] concerning the average distance property of compact connected metric spaces has interested many authors [2,6,7,8,11,13,14]. Gross and Stadje [10] independently proved the following remarkable result: The evaluation of the average distance constant of a given compact connected metric space is an important problem in numerical geometry.…”

Section: Introductionmentioning

confidence: 99%

Average distance constants for polygons in spaces with non-positive curvature

Kulshestha

Sag

Yang³

1990

Bull. Austral. Math. Soc.

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In this paper we show that the average distance constant of a general polygon which is a subset of an Af-space with non-positive curvature can be expressed as the extreme value of either of two nonlinear programs and discuss the practical application of one of these nonlinear programs for the determination of the average distance constant for a polygon in general, and in particular for a planar triangle.

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Numerical Geometry-Numbers for Shapes

Cited by 18 publications

References 0 publications

Rendezvous numbers of metric spaces – a potential theoretic approach

Rendezvous numbers of metric spaces – a potential theoretic approach

Minimax theorems with applications to convex metric spaces

Average distance constants for polygons in spaces with non-positive curvature

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