Let $\mathcal{H} = \lbrace u\in \mathbb{R}^n \mid \langle x, (1, 1, ...., 1) \rangle = 0 \rbrace $, the hyperplain where the avarage is zero i.e. $\frac{1}{n}\sum\limits_{i=1}^n x_i = 0$. Given two vectors $c,d \in \mathcal{H} $ and error bounds $e_c, e_d \in \mathbb{R}$ such that $\left \Vert c \right \Vert > e_c $ and $\left \Vert d \right \Vert > e_d $. Let $ x,y \in \mathcal{H} $ such that $\left \Vert x-c \right \Vert < e_c $ and $\left \Vert y-d \right \Vert < e_d $. What is the minimum and maximum angle between $x$ and $y$ $\frac{\langle x, y\rangle}{\left \Vert x \right \Vert \left \Vert y \right \Vert } $ ?

The range is preferably expressed with the angle between $c$ and $d$.


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