I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate any comments. Here is the formal statement:
Let $d \in \mathbb{N}$ and let $S^{(d-1)}$ denote the sphere in dimension $\mathbb{R}^d$. For a fixed $v \in S^{(d-1)}$ and $\theta \in [0,\pi/2)$, let $$ C_{\theta,v} = \{x \in S^{(d-1)}: \langle x,v \rangle \geq \cos(\theta) \}. $$
Conjecture Let $x_1$ and $x_2$ be two arbitrary elements of $C_{\theta,v}$. Is it correct that $$ \langle x_1,x_2 \rangle \geq \cos(2\theta). $$
