The question is kind of intuitive:
Consider a points $p$ in $\mathbb{R}^d$ ($d$-dimensional Euclidean space). $$p=\{x_1, x_2, \ldots, x_d\}$$
We can always transform it into spherical coordinate.
Denote the transformation by $M$, we have $q = M(p)$ and $$q = \{r, \alpha_1, \alpha_2, \ldots, \alpha_{d-1}\}$$ where $r$ is the radius and $\alpha_i$ are the angles.
Here is the problem:
Give 2 points $p_1$ and $p_2$, we can obtain is spherical coordinate $q_1 = M(p_1)$ and $q_2 = M(p_2)$. Assume that $q_1 = \{r, \alpha_1, \alpha_2, \ldots, \alpha_{d-1}\}$ and $q_2 = \{r, \beta_1, \beta_2, \ldots, \beta_{d-1}\}$ (i.e. they have same radius). In addition, $q_1$ and $q_2$ satisfy the following property: $$\alpha_i - \beta_i \leq \varepsilon ~~\forall i \in [1, d-1]$$
Can we find an upper bound for the angle $\theta$ between vector $\overrightarrow{op_1}$ and $\overrightarrow{op_2}$ where $o$ is the origin in $\mathbb{R}^d$ (in other words, $cos(\theta) = \frac{\overrightarrow{op_1} \cdot \overrightarrow{op_2}}{|\overrightarrow{op_1}| |\overrightarrow{op_2}|}$).
My conjecture is that $\theta \leq \sqrt{d-1} \varepsilon$, but it seems that this is not correct.
Any suggestions or comments?
Note that the bound should be applicable in higher dimension, say $d = 10$, not just 2 or 3 dimensional space.