Fekete points and the transfinite diameter of a compact set

Let $K\subseteq \mathbb{C}$ be compact and nonempty, let $n\ge 2$ be an integer, and write $K^n$ for the $n$-fold product with the product topology. For $z=(z_1,\dots ,z_n)\in K^n$ set

${\mathrm{\Delta }}_n(z):={\prod }_{1\le i<j\le n}(z_i-z_j),D_n(z):=\mathrm{\mid }{\mathrm{\Delta }}_n(z)\mathrm{\mid }={\prod }_{1\le i<j\le n}\mathrm{\mid }{z}_i-z_j\mathrm{\mid }\in [0,\mathrm{\infty })\mathrm{.}$

The formula for ${\mathrm{\Delta }}_n$ is the Vandermonde product, whose polynomial form in $n$ indeterminates is The Vandermonde polynomial ${\mathrm{\Delta }}_n={\prod }_{i<j}(x_i-x_j)$; only the numerical function $D_n$ on $K^n$ is used here. Each factor $(z_1,\dots ,z_n)\mapsto z_i-z_j$ is continuous: the projections are continuous for the product topology (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and complex subtraction is continuous since $\mathrm{\mid }(z-w)-(z_0-w_0)\mathrm{\mid }\le \mathrm{\mid }z-z_0\mathrm{\mid }+\mathrm{\mid }w-w_0\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive). Finite products and the modulus preserve continuity (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive); the product $K^n$ of finitely many compact spaces is compact (A product of finitely many compact spaces is compact in the product topology), and $K^n\mathrm{\ne }\mathrm{\varnothing }$. Hence $D_n$ and the composition

$(z_1,\dots ,z_n)⟼D_n(z{)}^{2\mathrm{/}[n(n-1)]}$

attain greatest values on $K^n$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value). The $n$-th Fekete diameter of $K$ is

${\delta }_n(K):={\max }_{(z_1,\dots ,z_n)\in K^n}{D}_n(z_1,\dots ,z_n{)}^{2\mathrm{/}[n(n-1)]}\in [0,\mathrm{\infty }),$

equivalently ${\delta }_n(K)=({\max }_{K^n}{D}_n{)}^{2\mathrm{/}[n(n-1)]}$, because $t\mapsto t^{2\mathrm{/}[n(n-1)]}$ is increasing on $[0,\mathrm{\infty })$ and the root is the nonnegative one (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$). The exponent $2\mathrm{/}[n(n-1)]=1\mathrm{/}\left(\genfrac{}{}{0px}{}{n}{2}\right)$ is the reciprocal of the number of unordered pairs, so that ${\delta }_n$ scales like a length. A tuple $z\in K^n$ at which the maximum is attained is an $n$-point Fekete tuple of $K$, and its entries $z_1,\dots ,z_n$ are $n$-point Fekete points.

To a Fekete tuple $z$ one associates its monic Fekete polynomial

$F_n(Z):={\prod }_{j=1}^n(Z-z_j),$

a monic polynomial of degree $n$ in the conventions of Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials: each factor $Z-z_j$ is monic of degree $1$, and degrees add and leading coefficients multiply under multiplication over the integral domain $\mathbb{C}$ (Over an integral domain, degrees add under multiplication of nonzero polynomials), by induction on $n$.

The transfinite diameter of $K$ is

$\tau (K):={\inf }_{n\ge 2}{\delta }_n(K)\in [0,\mathrm{\infty }),$

an infimum over a nonempty set of nonnegative reals, hence a well-defined real number (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)). For the empty set one uses the separate convention $\tau (\mathrm{\varnothing }):=0$. Whether the sequence $({\delta }_n(K){)}_{n\ge 2}$ is nonincreasing, and whether $\tau (K)$ is its limit, is not assumed in the definition.

Fekete tuples exist but are not unique. The maximum is attained by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, so every nonempty compact $K$ has at least one $n$-point Fekete tuple for every $n\ge 2$; no uniqueness is claimed, and no tuple is selected by the definition. The quantity $D_n$ is unchanged by permuting the entries, since a permutation permutes the factors $\mathrm{\mid }{z}_i-z_j\mathrm{\mid }$, so every permutation of a Fekete tuple is again one, and ${\delta }_n(K)$ is order-independent.

Sign and positivity. $D_n\ge 0$ always, and ${\delta }_n(K)>0$ holds exactly when $K$ has at least $n$ points: with $n$ distinct points of $K$ the product is positive, while a tuple with two equal entries has $D_n=0$. In particular ${\delta }_2(K)$ is the diameter of $K$, the exponent $2\mathrm{/}[2\cdot 1]=1$ recovering the unnormalized maximum of $\mathrm{\mid }{z}_1-z_2\mathrm{\mid }$.

Scaling. If $t\in \mathbb{C}\setminus \{0\}$ and $tK=\{tz:z\in K\}$, then $D_n(t{z}_1,\dots ,t{z}_n)=\mathrm{\mid }t{\mathrm{\mid }}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}{D}_n(z_1,\dots ,z_n)$ and ${\delta }_n(tK)=\mathrm{\mid }t\mathrm{\mid }{\delta }_n(K)$: the normalization by the number of pairs is exactly what makes the $n$-th Fekete diameter a length. Consequently $\tau (tK)=\mathrm{\mid }t\mathrm{\mid }\tau (K)$ as well.

Relation to the logarithmic capacity. The transfinite diameter is a purely combinatorial size functional, defined from extremal configurations of points, whereas the logarithmic capacity of Robin constant and logarithmic capacity of a compact set is defined variationally from a minimum-energy problem over probability measures. The definition here asserts no relation between the two; any comparison is proved later.

Choice. No choice principle is used: the extremal tuple is supplied by the extreme-value theorem for a continuous function on a nonempty compact space, the product of finitely many compact spaces is compact in ZF, and the infimum over $n\ge 2$ is a set-theoretic construction on a fixed set of reals.