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Statement

Quoted (p. 112): "Finally I shall deal with the following problem of Erdös, Herzog and Piranian [2, Problem 13]. Let $z_\nu$ be $n$ complex numbers which satisfy $|z_\mu-z_\nu|\le2$ ($\mu,\nu=1,\cdots,n$). Is $\prod_{\nu=1}\prod_{\mu\ne\nu}|z_\mu-z_\nu|$ maximal if the $z_\nu$ are the vertices of a regular $n$-gon of diameter 2? We denote the maximum by $\Delta_n$:

$$(10)\qquad \Delta_n=\max_{\substack{z_1,\cdots,z_n\\|z_\mu-z_\nu|\le2}} \prod_{\nu=1}^n\prod_{\mu\ne\nu}|z_\mu-z_\nu| .$$

The conjecture implies that $\Delta_n=n^n$ for even $n$ and $\Delta_n=n^n(\cos\pi/2n)^{-n(n-1)}$ for odd $n$. The last quantity is $n^n(1-\frac{\pi^2}{8n^2}+\cdots)^{-n(n-1)}\sim n^ne^{\pi^2/8}$."

Theorem 15 (p. 113). "Let $K$ be a convex continuum of capacity 1. Then, for $z_\nu\in K$ ($\nu=1,\cdots,n$), $\prod_{\nu=1}^n\prod_{\mu\ne\nu}|z_\mu-z_\nu|\le2^{4(n-1)}n^n$."

Theorem 16 (p. 114). "If $\Delta_n$ is defined by (10), then $\Delta_n\le2^{4(n-1)}\cdot n^n$."

Remark (pp. 114--115, quoted in part): "We can make the following observation in favor of the conjecture of Erdös, Herzog and Piranian about $\Delta_n$. The convex hull $K_n$ of a maximal system $\{z_n^{(n)},\cdots,z_n^{(n)}\}$ [sic] is nearly a disk, for large $n$."

The product in (10) is Problem 1045's $\Delta(z_1,\ldots,z_n)$ and the $D_k$ of Danzer and Pommerenke (1967); the even-$n$ value $n^n$ the conjecture implies was refuted there for every even $n\ge4$ (card), and this paper proves nothing about the regular polygon.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115; the problem and display (10) on printed p. 112, Lemma 6 on pp. 112--113, Theorem 15 on p. 113 with its proof on pp. 113--114, Theorem 16 with its proof and the remark on pp. 114--115 (PDF pp. 16--19 of the publisher's scan), read on the page images (the scan has no text layer). The copy read is identified in the source digest.

Read depth. Claims checked: the problem as recalled, display (10), Lemma 6, Theorems 15 and 16 and the remark were read clause by clause on the page images on 2026-09-22; the proofs of Theorems 15 and 16 and of the remark (half a page together) were read in full and followed, Lemma 6 and Szegő's determinant inequality being taken as printed. The proof of Lemma 6 (p. 113) was read for structure and not checked. Nothing here is independently reviewed.

Proof pointer

Lemma 6 (pp. 112--113): for a convex continuum $K$ of capacity 1 there are monic polynomials $f_n(z)=z^n+\cdots$, $n=1,2,\ldots$, with zeros in $K$ and $\max_{z\in K}|f_n(z)|\le4$; the zeros are $\psi(e^{2\pi i\nu/n})$ for the exterior map $\psi(w)=w+\cdots$ of $K$, and for fixed $z\in K$ the function $\Phi(w)=e^{-\pi i(n+1)/n}\prod_\nu(\psi(e^{2\pi i\nu/n}w)-z)^{1/n}$ is starlike and univalent in $|w|>1$ (convexity of $K$ makes each factor starlike about $0$), so $\Psi(w)=\Phi(w^{1/n})^n=w+\cdots$ is univalent and nonvanishing in $|w|>1$, $\max_{|w|=1}|\Psi|\le4$, and $|f_n(z)|=|\Psi(1)|\le4$.

Theorem 15 (pp. 113--114): $\prod_\nu\prod_{\mu\ne\nu}|z_\mu-z_\nu|$ is the squared modulus of the Vandermonde determinant $\det(z_j^{k-1})$, which equals $\det(f_{k-1}(z_j))$ for any monic $f_k$ of degree $k$ ($f_0=1$); Hadamard's determinant inequality gives $n^n\max_K|f_1|^2\cdots\max_K|f_{n-1}|^2$ (an inequality the paper attributes to Szegő, citing footnote 7 on p. 236 of its [3]), and Lemma 6 bounds each factor by $16$, giving $4^{2(n-1)}n^n$.

Theorem 16 (p. 114): for a maximal system with convex hull $K$, Theorem 15 applied after scaling $K$ to capacity 1 gives $(12)\ \Delta_n\le2^{4(n-1)}n^n(\operatorname{cap}K)^{n(n-1)}$, and $2\operatorname{cap}K\le\operatorname{diam}K\le2$ gives $\operatorname{cap}K\le1$. Remark (pp. 114--115): if the hulls $K_n$ were not nearly disks, a subsequence would converge to a convex $K_0$ of diameter at most 2 that is not a disk, so $\operatorname{cap}K_0<1$ and $\operatorname{cap}K_{n_k}\le1-\delta<1$, and (12) would give $\Delta_{n_k}\le2^{4(n_k-1)}n_k^{n_k}(1-\delta)^{n_k(n_k-1)}<1$ for large $k$, which contradicts $\Delta_{n_k}\ge n_k^{n_k}$.

Dependencies

Within the paper: Lemma 6. Outside it: Szegő's form of Hadamard's determinant inequality (Fekete, Math. Z. 17 (1923), 228--249, footnote 7 on p. 236, the paper's [3]; not held), the bound $\max_{|w|=1}|\Psi|\le4$ for a univalent nonvanishing $\Psi(w)=w+\cdots$ in $|w|>1$, and $2\operatorname{cap}K\le\operatorname{diam}K$.

Bears on

• Problem 1045: the first general upper bound $\Delta_n\le2^{4(n-1)}n^n$ on the problem's maximum, exponentially above the conjectured $n^n$ scale, later replaced by the $n^n\exp(15n^{6/7})$ of Danzer and Pommerenke (1967); the remark shows the convex hull of a maximal system is nearly a disk for large $n$. The paper decides nothing about the regular polygon, and its record of the conjecture's even value $n^n$ is the statement the 1967 paper refutes.