A spherical cap is defined by $C(x_1,\alpha_1)=\{y \in S^{n-1} : x\cdot y \geq \cos(\alpha)\}$ and $\alpha_i\in [0, \pi]$ and $x_i \in S^{n-1}$ ( $x_1 \cdot x_2$ refers to the inner product of $x_1,x_2$)
I would like to show: $ C(x_1,\alpha_1)^\circ \cap C(x_2,\alpha_2)^\circ\not = \emptyset $ if and only if $ x_1\cdot x_2 \in (\cos(\alpha_1+\alpha_2),1] $
"$\Longrightarrow$ " if $ C(x_1,\alpha_1)^\circ \cap C(x_2,\alpha_2)^\circ\not = \emptyset $ then $b\in S^{n-1}$ exists with $b\cdot x_1 \geq \cos(\alpha_1)$ and $ b\cdot x_2 \geq \cos(\alpha_2)$ $\ldots$
does anybody have an idea that could help me?