---

Claim. Stijn Cambie, Arne Decadt, Yanni Dong, Tao Hu and Quanyu Tang, On the maximum product of distances of diameter $2$ point sets (arXiv:2603.07088), normalize the product of Problem 1045 as $\overline\Delta=\Delta/n^n$. Proposition 13 gives $\overline\Delta_{\max}(n)=1$ for $n\le2$ and $\overline\Delta_{\max}(3)=64/27$; Proposition 14 gives $\overline\Delta_{\max}(4)=16(7-4\sqrt3)$, attained only at the kite $\{0,2,\sqrt3+i,\sqrt3-i\}$ up to congruence and relabeling; Proposition 15 gives $\overline\Delta_{\max}(5)=(4/5)^5(\sqrt5-1)^{10}$, attained only at the regular pentagon. The paper also proves structural conditions on maximizers and even-order constructions with $\liminf\overline\Delta_{\max}(n)\ge C\approx1.268$ along even $n$; its Theorem 3 was first posted in Tang's note of 18 December 2025. The results are recorded on the source card.

Covers. The maximum and the maximizer for every $n\le5$: the regular polygon is the unique maximizer at $n=5$ and is not one at $n=4$. The values for $n\le4$ were already determined by Danzer and Pommerenke; the case $n=5$ is new.

Depends on. No page of this wiki.

Standing. An arXiv preprint, first posted on 7 March 2026 and linked from the site's thread on 10 March 2026, with no journal publication recorded. The site's commentary credits the paper, but the site labels the problem OPEN, so the credit is not review.