Maximize $\prod_{i\ne j}|z_i-z_j|$ under diameter $\le 2$: are regular polygons optimal for odd $n$? (Erdős #1045)
Statement
Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert z_i-z_j\rvert\leq 2$ for all $i,j$, and set $$\Delta(z_1,\ldots,z_n)=\prod_{i\neq j}\lvert z_i-z_j\rvert.$$ What is the maximum possible value of $\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon (scaled to have diameter $2$)? It is now known that the regular polygon is NOT the maximiser for any even $n\geq 4$ (see background); the question remains open for odd $n$, where the regular polygon attains $$\Delta=\cos(\pi/2n)^{-n(n-1)}\,n^n \sim e^{\pi^2/8}\,n^n.$$
Acceptance. FULLY RESOLVES: a proof that for every odd $n\geq 3$ the regular $n$-gon of diameter $2$ maximises $\Delta$ among all diameter-$\leq 2$ configurations (machine-checkable preferred, else a complete written proof); OR a counterexample at some odd $n$ — an explicit configuration $z_1,\ldots,z_n$ (exact algebraic data or certified enclosures, not bare floats) with a rigorous interval-arithmetic certificate that all pairwise distances are $\leq 2$ and that $\Delta$ strictly exceeds $\cos(\pi/2n)^{-n(n-1)}n^n$; OR a full determination of $\max\Delta$ as a function of $n$. ADVANCES: certified configurations improving the even-$n$ lower-bound constant strictly beyond the best constant stated in the background; determination of $\lim(\max\Delta)/n^n$ for even $n$ or for odd $n$; a certified globally optimal configuration for some specific $n\geq 4$ (configuration plus a matching rigorous upper bound, e.g. from a converging SDP/moment relaxation); or structural theorems about maximisers strictly extending those of [CDDHT26]. Deliver the configurations with certification code (reproducible), or the proof file.
Background
Posed by Erdős, Herzog, and Piranian [EHP58, p.143]; listed as open on erdosproblems.com/1045 (fetched 2026-07-13, status 'open', tagged 'analysis'). [EHP58] proved that if $f$ is a monic polynomial with roots $z_1,\ldots,z_n$ and $\{z:\lvert f(z)\rvert<1\}$ is connected, then $\prod_{i\neq j}\lvert z_i-z_j\rvert<n^n$. For regular polygons of diameter $2$ the value of $\Delta$ is $n^n$ when $n$ is even and $\cos(\pi/2n)^{-n(n-1)}n^n\sim e^{\pi^2/8}n^n\approx 3.433\,n^n$ when $n$ is odd. Pommerenke [Po61] proved the general upper bound $\Delta\leq 2^{O(n)}n^n$ for all configurations of diameter $\leq 2$. On the extremal side the picture has moved recently: Hu and Tang (site comments) found configurations beating the regular polygon for $n=4$ and $n=6$, and Cambie (site comments) showed the regular polygon is non-optimal for every even $n\geq 4$. The current frontier is in Sothanaphan [So25] and Cambie–Decadt–Dong–Hu–Tang [CDDHT26]: for even $n$, $\liminf (\max\Delta)/n^n\geq C$ with best known $C\approx 1.268$, improved to $C\approx 1.304$ along $n$ divisible by $6$; [CDDHT26] also proves structural results about extremal configurations and explicitly conjectures that the regular polygon IS the maximiser for all odd $n$. Whether $\lim(\max\Delta)/n^n=e^{\pi^2/8}$ for odd $n$ is unknown. The attacker's tool: high-precision numerical optimization over planar point configurations (gradient methods, basin hopping, symmetry ansätze) at concrete $n$, with rigorous interval-arithmetic certification of the diameter constraint and of record $\Delta$ values — either beating the regular polygon at some odd $n$, improving the even-$n$ constants, or certifying global optimality at small $n$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1045 (T. F. Bloom) | website |
Investigations · 0
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