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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?

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If the length of the great-circle path from $x_1$ to $x_2$ is less than $\alpha_1+\alpha_2$, there should be a point $b$ on the path whose distance along the path from $x_1$ is less than $\alpha_1$ and whose distance from $x_2$ is less than $\alpha_2$.

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