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...

This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant c n (X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that c n (X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then c n (H ) = n n/2 . The same value of c n (H ) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order d n/2 , for n large enough, and also an integral form of the asymptotic quantity c(H ), that is the (linear) polarization constant of the Hilbert space H , where dim H = d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of c n (H ) and c(H ). In particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999) ✩

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We extend some equilibrium‐type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T≃[0,2π), but the motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton. The problem is to minimize — with respect to the arbitrary translates y0=0,yj∈T, j=1,⋯,n — the maximum of the sum function 0trueF:=K0+∑j=1nKj(·−yj), where the functions Kj are certain fixed ‘kernel functions'. In our setting, the function F has singularities at functions yj, while in between these nodes it still behaves regularly. So one can consider the maxima mi on each subinterval between the nodes yj, and minimize maxF=trueprefixmaximi. Also the dual question of maximization of trueprefixminimi arises. Hardin, Kendall and Saff considered one even kernel, Kj=K for j=0,⋯,n, and Fenton considered the case of the interval [−1,1] with two fixed kernels K0=J and Kj=K for j=1,⋯,n. Here we build up a systematic treatment when all the kernel functions can be different without assuming them to be even. As an application we generalize a result of Bojanov about Chebyshev‐type polynomials with prescribed zero order.