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Monotonicity of the Fekete diameters and the transfinite diameter

Statement

Let $K \subseteq C$ be nonempty and compact, with the Fekete diameters $δ_{n} (K)$ and transfinite diameter $τ (K)$ of Fekete points and the transfinite diameter of a compact set. Then

$δ_{n + 1} (K) \leq δ_{n} (K) (n \geq 2),$

the sequence $(δ_{n} (K))_{n \geq 2}$ therefore converges, and

$τ (K) = \inf_{n \geq 2} δ_{n} (K) = \lim_{n \to \infty} δ_{n} (K) .$

No choice principle is used.

Facts & Assumptions

Given: a nonempty compact $K \subseteq C$ and the quantities $D_{n}$ , $Δ_{n}$ , $δ_{n} (K)$ , $τ (K)$ and Fekete tuples of Fekete points and the transfinite diameter of a compact set.

[F1]

For $n \geq 2$ and $z = (z_{1}, \dots, z_{n}) \in K^{n}$ one has $Δ_{n} (z) = \prod_{i < j} (z_{i} - z_{j})$ and $D_{n} (z) = ∣ Δ_{n} (z) ∣ = \prod_{i < j} ∣ z_{i} - z_{j} ∣ \in [0, \infty)$ ; the maximum of $D_{n}$ over the nonempty compact $K^{n}$ is attained and $δ_{n} (K) = (\max_{K^{n}} D_{n})^{2 / [n (n - 1)]} \in [0, \infty)$ ; and $τ (K) = \inf_{n \geq 2} δ_{n} (K)$ (Fekete points and the transfinite diameter of a compact set). In particular $D_{n} (z) \leq δ_{n} (K)^{(\binom{n}{2})}$ for every $z \in K^{n}$ , because $t \mapsto t^{2 / [n (n - 1)]}$ is increasing on $[0, \infty)$ and $1 / [n (n - 1)] \cdot 2 = 2 / [n (n - 1)]$ is the reciprocal of $(\binom{n}{2}) = \frac{n (n - 1)}{2}$ .

[F2]

$∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z ∣ \geq 0$ for all $z, w \in C$ , and $∣ z ∣ = 0$ exactly for $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F3]

Every $a \geq 0$ has a unique $n$ -th root $a^{1 / n} \geq 0$ , the map $t \mapsto t^{n}$ is strictly increasing on $[0, \infty)$ for $n \geq 1$ , and for $x, y \geq 0$ one has $x^{1 / n} \leq y^{1 / n}$ exactly when $x \leq y$ (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ , Monotonicity of $x \mapsto x^{n}$ and of $n \mapsto a^{n}$ ).

[F4]

A monotone sequence of reals converges if and only if it is bounded (A monotone sequence converges if and only if it is bounded).

[F5]

If a sequence of reals converges to $L$ and $x_{k} \leq c$ for all $k$ , then $L \leq c$ ; if $x_{k} \geq c$ for all $k$ , then $L \geq c$ (Limits preserve non-strict inequalities).

[F6]

If $S \subseteq R$ is nonempty and bounded below, then $\inf S$ is a lower bound of $S$ and no larger lower bound exists; in particular a lower bound $ℓ$ of $S$ equals $\inf S$ when every lower bound is $\leq ℓ$ (Greatest lower bound (infimum), Every nonempty set bounded below has an infimum).

Proof

technique · direct

1.1F1given

Fix $n \geq 2$ and a tuple $z = (z_{1}, \dots, z_{n + 1}) \in K^{n + 1}$ . For $k \in {1, \dots, n + 1}$ let $A_{k}$ be the $n$ -tuple obtained by deleting the $k$ -th entry of $z$ . Each $A_{k}$ lies in $K^{n}$ , so by [F1] $D_{n} (A_{k})$ is at most $δ_{n} (K)^{(\binom{n}{2})}$ ; that is, $D_{n} (A_{k}) \leq δ_{n} (K)^{n (n - 1) / 2}$ .

2.1step 1.1F1F2

Expanding the factors by [F1] and [F2] gives $\prod_{k = 1}^{n + 1} D_{n} (A_{k}) = \prod_{k = 1}^{n + 1} \prod_{i < j, i, j \neq k} ∣ z_{i} - z_{j} ∣$ ; the unordered pair ${i, j}$ contributes the factor $∣ z_{i} - z_{j} ∣$ for exactly those $k \notin {i, j}$ , that is for $n - 1$ of the $n + 1$ indices $k$ , so the last product equals $\prod_{i < j} ∣ z_{i} - z_{j} ∣^{n - 1} = D_{n + 1} (z)^{n - 1}$ , the last equality again by [F1] and [F2].

3.1step 2.1F1F3algebra

Multiplying the $n + 1$ inequalities of step 1.1 and substituting step 2.1 yields $D_{n + 1} (z)^{n - 1} \leq δ_{n} (K)^{(n + 1) n (n - 1) / 2}$ for every $z \in K^{n + 1}$ . Taking the maximum over $z \in K^{n + 1}$ and using that $t \mapsto t^{n - 1}$ is increasing on $[0, \infty)$ together with [F3] gives $δ_{n + 1} (K)^{(n + 1) n (n - 1) / 2} \leq δ_{n} (K)^{(n + 1) n (n - 1) / 2}$ , since $(\binom{n + 1}{2}) (n - 1) = \frac{(n + 1) n (n - 1)}{2}$ is the exponent obtained from $D_{n + 1} (z) \leq δ_{n + 1} (K)^{(\binom{n + 1}{2})}$ . The common exponent $E : = \frac{(n + 1) n (n - 1)}{2}$ is a positive integer (as $n \geq 2$ ), so [F3] applied to the two nonnegative numbers $δ_{n + 1} (K)$ and $δ_{n} (K)$ gives $δ_{n + 1} (K) \leq δ_{n} (K)$ .

4.1step 3.1F1F4F5F6

The sequence $(δ_{n} (K))_{n \geq 2}$ is nonincreasing by step 3.1 and bounded below by $0$ because $δ_{n} (K) \geq 0$ by [F1]; hence it converges, with limit $L \in R$ , by [F4]. Since $δ_{m} (K) \leq δ_{n} (K)$ for all $m \geq n \geq 2$ , [F5] applied to the tail from $n$ gives $L = \lim_{m} δ_{m} (K) \leq δ_{n} (K)$ for every $n \geq 2$ , so $L$ is a lower bound of ${δ_{n} (K) : n \geq 2}$ ; and if $ℓ$ is any lower bound of that set, then $δ_{n} (K) \geq ℓ$ for every $n$ and [F5] gives $L \geq ℓ$ . Thus $L$ is the greatest lower bound and $L = \inf_{n \geq 2} δ_{n} (K) = τ (K)$ by [F6] and [F1].

5.1step 3.1step 4.1∎

Combining steps 3.1 and 4.1, $δ_{n + 1} (K) \leq δ_{n} (K)$ for every $n \geq 2$ and $τ (K) = \lim_{n \to \infty} δ_{n} (K)$ , which is the statement.

Remarks

Where the normalization enters. The exponent $2 / [n (n - 1)]$ in the definition of $δ_{n}$ is exactly what makes the exponents on the two sides of step 3.1 agree: the pair-count $(\binom{n + 1}{2}) (n - 1)$ of the $(n + 1)$ -tuple side equals the pair-count $(\binom{n}{2}) (n + 1)$ of the $n$ -tuple side, both equal to $\frac{(n + 1) n (n - 1)}{2}$ .

Choice. The argument uses only real algebra and order-completeness facts; no choice principle is involved, and the extremal tuples are maxima of continuous functions on compact product spaces supplied by Fekete points and the transfinite diameter of a compact set.

Depends on

Used by

Fekete–Szegő equality of logarithmic capacity, transfinite diameter, and Chebyshev constant Theorem

Dependency tree · two levels

55 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. B. Saff, Logarithmic Potential Theory with Applications to Approximation Theory, §1 (standard reference, not scraped)
B. Khoruzhenko, LTCC Potential Theory notes, §5 (standard reference, not scraped)