Problem 1045

Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$ , and

\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i-z_j\rvert.

What is the maximum possible value of $\Delta$ ? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?

Source: erdosproblems.com/1045

Analysis

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label (page last edited 02 April 2026). Danzer and Pommerenke's accepted partial claim determines the maximum for $n\le4$ and answers the regular-polygon question negatively for every even $n\ge4$ . Pending partial claims determine the maximum for $n=5$ and $n=6$ and, in Boyang Hu's manuscript, the maximizer for every odd $n\ge2^{10^8}$ and every even $n\ge2^{10^{120}}$ . None covers every $n$ , so the standing stays open.

CDDHT26: S. Cambie, A. Decadt, Y. Dong, T. Hu, and Q. Tang, On the maximum product of distances of diameter 2 point sets. arXiv:2603.07088 (2026).
DP67: L. Danzer and Ch. Pommerenke, Über die Diskriminante von Mengen gegebenen Durchmessers. Monatsh. Math. 71 (1967), 100-113.
EHP58: Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
Po61: Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; the problem's product as display (10), p. 112; Theorem 15, p. 113; Theorem 16 and the remark on the hull of a maximal system, pp. 114--115. Library home: Pommerenke 1961 (result page Pommerenke 1961, Theorem 16).
So25: N. Sothanaphan, An improved lower bound to Erdos' problem concerning products of distances for fixed diameter. arXiv:2512.14251 (2025).

Problem 1045

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Proof claims

Discussion