$|\langle a_i, a_j\rangle|$ for $p$ points on a unit circle.

Question

Is it true that given any $p$ points $a_1, .., a_p$ on a unit [euclidean] circle, there is always a pair $i \ne j$ such that $|\langle a_i, a_j\rangle| \ge \cos{\pi/p}$?

Answer 1

Of the $2p$ points $\pm a_1,\ldots,\pm a_p$, there must be two adjacent points whose angular separation is $\le \frac{\pi}{p}$.