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Claim. Theorem 1 of L. Danzer and Ch. Pommerenke, Über die Diskriminante von Mengen gegebenen Durchmessers, Monatsh. Math. 71 (1967), 100--113, works with the ordered product $D_k$ of Problem 1045 maximized over $k$-point sets of diameter at most $2$. It proves

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

so the regular polygon is optimal at $k=2,3$ and the kite $\{0,2,\sqrt3+i,\sqrt3-i\}$ beats the square at $k=4$. It also proves $D_k/k^k>1+\frac{\pi^4}{32k}\bigl(1-\frac5{2k}-\frac2{k^2}\bigr)$ for $k\equiv2\pmod4$, $k\ge6$, and $D_k/k^k>1+\frac{\pi^4}{32k}\bigl(1-\frac4k-\frac6{k^2}\bigr)$ for $k\equiv0\pmod4$, $k\ge8$, through an alternating-radius construction. With $D_4$ this gives $D_k>k^k$ for every even $k\ge4$, while the regular even $k$-gon of diameter $2$ has product $k^k$. Theorem 2 gives $D_k<k^k\exp(15k^{6/7})$. The results are recorded on the source card.

Covers. The maximum for $n=2,3,4$ (the regular polygon at $n=2,3$, the kite at $n=4$), and a negative answer to the regular-polygon question for every even $n\ge4$. It does not determine the maximum for $n\ge5$ or decide the regular-polygon question for odd $n\ge5$.

Depends on. No page of this wiki.

Acceptance. Refereed: Monatshefte für Mathematik 71 (1967), 100--113, whose issue the publisher's record dates April 1967; the page name uses the first day of that month. The site labels the problem OPEN.