A conjecture about random angles formed between n points
Posted: Mon Jul 19, 2021 4:32 am
Let $\alpha(n)$ be the smallest non-negative angle such that for any $n\geq3$ distinct points in the plane, there exists points $A,B,C$ such that $\angle ABC\leq\alpha(n)$.
My conjecture is $\alpha(n)=\frac{\pi}{n}$ for all $n\geq3$.
If the points are in convex configuration, it's easy to prove by Pigeonhole Principle. Also $n=3,4,5,6$ cases are doable in every configuration. I could not go any further. But is this true for all $n$ and every configuration? Or is there a counter example to disprove it?
My conjecture is $\alpha(n)=\frac{\pi}{n}$ for all $n\geq3$.
If the points are in convex configuration, it's easy to prove by Pigeonhole Principle. Also $n=3,4,5,6$ cases are doable in every configuration. I could not go any further. But is this true for all $n$ and every configuration? Or is there a counter example to disprove it?