Problem 1045

Erdős Problem #1045: declared status 'open'. Formalized: no. 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? Current best: Cambie (also in the comments) showed that, in general, the vertices of a regular polygon are not a maximiser for all even $n\geq 4$. Sothanaphan, An improved lower bound to Erdos' problem concerning products of distances for fixed diameter. Prize: no. Tags: analysis.