# Angle between two centered noisy vectors

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 $d$.