Potential theoretic approach to rendezvous numbers

Details Potential theoretic approach to rendezvous numbers Potential theoretic approach to rendezvous numbers

2005-03-21

arxiv.org/abs/math/0503423v2

Monatshefte fur Mathematik 148 (2006), 309-331

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

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 {x1,...,xn} in X, there exists some point x in X with the average of the distances d(x,xj) 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.

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