Rendezvous numbers of metric spaces – a potential theoretic approach

Farkas, Bálint; Révész, Szilárd Gy.

doi:10.1007/s00013-005-1513-9

Cited by 12 publications

(16 citation statements)

References 22 publications

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“…Also the Wiener energy w(K) has connection to invariant measures, as shown by the following result, which is a simplified version of a more general statement from [10], see also Wolf [26].…”

Section: Average Distance Number and The Maximum Principlementioning

confidence: 94%

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Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces

Farkas

Nagy

2008

Potential Anal

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We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel satisfies the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the Chebyshev constant of any compact set coincides with its transfinite diameter, the kernel must satisfy the maximum principle. An abundance of examples is provided to show the sharpness of the results.Keywords Transfinite diameter · Chebyshev constant · Energy · Potential theoretic kernel function in the sense of Fuglede · Frostman's maximum principle · Rendezvous and average distance numbers Mathematics Subject Classifications (2000) 31C15 · 28A12 · 54D45

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Section: Average Distance Number and The Maximum Principlementioning

confidence: 94%

“…It was proved in [10] that one always has M(K) = r(K), so once we have an invariant measure, then the Chebyshev constant is again easy to determine.…”

Section: Average Distance Number and The Maximum Principlementioning

confidence: 97%

Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces

Farkas

Nagy

2008

Potential Anal

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“…-can all be conveniently described by potential theory, hence it is natural to expect general versions of the results known so far. For these see [9].…”

Section: Concluding Remarks and Hints Of Further Workmentioning

confidence: 99%

Potential Theoretic Approach to Rendezvous Numbers

Farkas

Révész

2006

Mh Math

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We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X, d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x 1 , . . . , x n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x, x j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named "the magic number" of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.AMS Subj. Clas.(2000): Primary: 31C15. Secondary: 28A12, 54D45 Keywords 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, transfinite diameter, Chebyshev constant, weak * -topology, (weak) rendezvous number, average distance, minimax theorem.by the topic. They calculated the rendezvous numbers in particular cases, and extended the results in the direction of weak rendezvous numbers or rendezvous numbers of unit spheres in Banach spaces (see, e.g., [2], [4], [7], [16], [21], [22], [24], [35], [36], [37,38,39] and [40]).Already Björck applied certain tools of potential theory in studying constants related to rendezvous numbers [4]. Now, exploring notions of general, abstract potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka -and, in particular, drawing from some exploration of mutual energies, Chebyshev constants and transfinite diameters over locally compact topological spaces with lower semicontinuous, nonnegative and symmetric kernels (see [15], [26], [8]) -we arrive at an understanding of these quantities from a more general viewpoint.However, to achieve this, we need to recover and even partially extend the relevant basic material. In particular, we thoroughly investigate energies and Chebyshev constants, and even also their "minimax duals" in function of two sets. The technical reason for that is that the classical definitions are kind of saddle point special cases of these more general notions, and we need to utilize special monotonicity and other properties, which stay hidden when considering only the diagonal cases.Let us recall the appropriate setting of potential theory in locally compact spaces. First, +∞ is added to the set of real numbers, i.e., we let R := R ∪ {+∞} endowed with its natural topology such that R + will be compact. Moreover, we will use the notation conv E for the convex hull of a subset E ⊂ R and conv E for the closed convex hull in R + , meaning, for example, conv(0, +∞) = [0, +∞].Throughout the paper X denotes a locally compact Hausdorff space, and k : X × X → R is a kernel function in the sense o...

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“…In [11] the background to our work, and in particular related work by other authors (see [1,4,7,8,15], for example), was discussed in some detail, and this discussion will not be repeated here.…”

Section: Introductionmentioning

confidence: 98%

Finite quasihypermetric spaces

Nickolas

Wolf

2009

Acta Math Hung

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Let (X, d) be a compact metric space and let M(X) denote the space of all nite signed Borel measures on X. Dene I : M(X) → R by I(μ) = X X d(x, y) dμ(x)dμ(y), and set M (X) = sup I(μ), where μ ranges over the collection of measures in M(X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) 0 for all measures μ in M(X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure.This paper explores the constant M (X) and other geometric aspects of X in the case when the space X is nite, focusing rst on the signicance of the maximal strictly quasihypermetric subspaces of a given nite quasihypermetric space and second on the class of nite metric spaces which are L 1 -embeddable.While most of the results are for nite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].

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Rendezvous numbers of metric spaces – a potential theoretic approach

Cited by 12 publications

References 22 publications

Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces

Transfinite Diameter, Chebyshev Constant and Energy on Locally Compact Spaces

Potential Theoretic Approach to Rendezvous Numbers

Finite quasihypermetric spaces

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