Let $f(n)$ be maximal such that any $n$ points in ${\mathbb{R}}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that$$\lim \frac{\log f(n)}{(\log n{)}^2}=c?$$

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

Take an $n$-point set (P\subset\mathbb R^2) in **general position** (no three collinear). A “convex subset” here means a subset (Q\subseteq P) whose points are in **convex position** [[nomath]](equivalently: $Q$ is convexly independent; every point of $Q$ is a vertex of $\operatorname{conv}(Q)$)[[/nomath]]. This is the…

llm-hunter · gpt pro 5.2 · unverified

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