Problem 1045

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claims/: The 5 claim pages of Problem 1045, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Status. 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.

Source. erdosproblems.com/1045, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1045, https://www.erdosproblems.com/1045.

References.

• [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_metric_properties_complex_polynomials (result page theorem_16).
• [So25] N. Sothanaphan, An improved lower bound to Erdos' problem concerning products of distances for fixed diameter. arXiv:2512.14251 (2025).

Formalization. The community database listed no formal statement for the problem on 2026-09-04. Two Lean developments exist, each linked at its pinned revision on its claim page: coleski's six-point repository and the repository accompanying Boyang Hu's manuscript. This corpus has built or audited neither.

Current assessment

The extremal-value problem remains open, but the regular-polygon question is already false for every even $n\geq4$. Erdős, Herzog, and Piranian posed the diameter-constrained discriminant problem in 1958. Danzer and Pommerenke then gave explicit even-order improvements over the regular polygon in 1967 and determined the exact values for $n\leq4$; Erdős's 1976 survey records the even-order disproof while retaining the regular polygon as the expected odd-order optimizer. See Erdős--Herzog--Piranian 1958, Danzer--Pommerenke 1967, and Erdős 1976.

For the ordered product used here, Danzer and Pommerenke proved

$$D_2=4,\qquad D_3=64,\qquad D_4=4096(7-4\sqrt3),$$

and $D_n>n^n$ for every even $n\geq4$, whereas the diameter-$2$ regular even $n$-gon has product $n^n$. Their alternating-radius construction proves the strict inequality for every even $n\ge6$ (at $n=4$ it falls below $4^4$, printed p. 107, footnote 4), and their exact value of $D_4$ gives it for $n=4$; their general upper bound gives $D_n<n^n\exp(15n^{6/7})$ for sufficiently large $n$. They do not determine $D_n$ in general or settle regular-polygon optimality for odd $n\geq5$. The result is recorded as an accepted partial claim.

Before that, Pommerenke's 1961 paper stated the same ordered product as $\Delta_n$ (display (10), p. 112), proved the first general upper bound $\Delta_n\leq 2^{4(n-1)}n^n$ (Theorem 16, p. 114, from Theorem 15 on points in a convex continuum of capacity 1), and remarked that the convex hull of a maximal system is nearly a disk for large $n$ (pp. 114--115); it decides nothing about the regular polygon. See Pommerenke 1961, Theorem 16.

Recent work improves lower bounds and resolves another small case without closing that gap. Cambie, Decadt, Dong, Hu, and Tang prove the exact optimum through $n=5$, report computational candidates at higher orders, and obtain new constructions and general estimates; Sothanaphan gives an asymptotic lower-bound improvement. The logarithmic-energy literature provides useful fixed-support asymptotics, but those results do not by themselves optimize the support together with the points as E1045 requires. See Cambie et al. 2026, Sothanaphan 2025, and Brauchart 2024.

The exact maxima for $n\le5$ of Cambie, Decadt, Dong, Hu, and Tang are recorded as a pending partial claim, and Hu and Tang's note of 3 October 2025 with counterexamples at $n=4$ and $n=6$ as another. Sothanaphan's arXiv:2512.14251v1 (16 December 2025) has no claim page: its lower bound $\liminf\Delta_{\max}(n)/n^n\ge1.0378$ along even $n$ settles no instance that Danzer and Pommerenke had not settled in 1967, and the 2026 paper improves its constant. Cambie's argument of 3 October 2025 that the regular polygon fails for every even $n\ge4$, which the site credits, has no claim page either: it is a thread post without a manuscript, and Danzer and Pommerenke's theorem covers it.

The repository github.com/coleski/erdos1045-n6 (11 September 2026, developed using Codex, and reference [10] of Hu's manuscript) claims the exact six-point maximum $64(2\sqrt3-2)^{18}$ with a Lean proof that this corpus has not built. It is recorded as a pending partial claim.

Proof claim on the site. The site's proof-claims tab carries a claim, filed as full, by Boyang Hu under the username Rogerhu (initial draft generated with GPT-6 Pro and a Lean development built with Astra, as the claim's notes say), submitted 2026-09-23 with a manuscript and a Lean repository: for every odd $n\ge2^{10^8}$ the regular polygon is the unique maximizer, and for every even $n\ge2^{10^{120}}$ the maximizer is unique and its diameter graph is a cycle on $n-3$ vertices with three pendant edges. The one thread comment notes that the problem asks for every $n$. The site labels the problem OPEN (page last edited 02 April 2026), and the claim is recorded as a partial claim on its page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• armentano_et_al_2025_characterization_logarithmic_fekete_critical_configurations_at_most_six_points_all_dimensions
• brauchart_2024_complete_minimal_logarithmic_energy_asymptotics_points_compact_interval_consequence_discriminant_ja
• cambie_2026_maximum_product_distances_diameter_point_sets
• danzer_pommerenke_1967_ber_die_diskriminante_von_mengen_gegebenen_durchmessers
• danzer_pommerenke_1967_ber_die_diskriminante_von_mengen_gegebenen_durchmessers / remark_p101
• danzer_pommerenke_1967_ber_die_diskriminante_von_mengen_gegebenen_durchmessers / theorem_1
• danzer_pommerenke_1967_ber_die_diskriminante_von_mengen_gegebenen_durchmessers / theorem_2
• erdos_1976_extremal_problems_polynomials
• erdos_1976_extremal_problems_polynomials / conjecture_p350
• pommerenke_1961_metric_properties_complex_polynomials
• pommerenke_1961_metric_properties_complex_polynomials / theorem_16
• sothanaphan_2025_improved_lower_bound_erdos_problem_concerning
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_13
• erdos_1958_metric_properties_polynomials / theorem_10