Let $z_1,\dots ,z_n\in \mathbb{C}$ with $|z_i-z_j|\le 2$ for all $i,j$, and$$\Delta (z_1,\dots ,z_n)=\prod _{i\ne j}|z_i-z_j|.$$What is the maximum possible value of $\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This question (in essentially this exact form) goes back to Erdős–Herzog–Piranian, and as of late 2025 the **exact maximum is not known in general**. What *is* known is:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.