Potential Theoretic Approach to Rendezvous Numbers

Abstract

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...

Rendezvous numbers of metric spaces – a potential theoretic approach

“…we see that R n (H, L), R(H, L) and A(H, L) are all of the form µ A(µ, H), with µ ranging over all averages of n Dirac measures at points of H, over all measures finitely supported in H and having only rational probabilities, and over all of M 1 (H), respectively, see [6].…”

“…Again, for good reasons (explained in more detail in [6]) we define these notions in dependence of two sets as variables. Definition 1.6.…”

“…The following proposition is proved by showing that the respective sequences are quasi-monotonous, hence Fekete's lemma (see [8], or also [5], [6]) applies.…”

“…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] with the study of related notions such as invariant measures and maximal energy (Wolf [23], Björck [3]). It turns out that general principles such as existence of capacitary measures or Frostman's equilibrium theorem are accounted for the existence of invariant measures.…”

“…, 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.,…”