bound the norm of the sum of the difference between the orthogonal projection of two vectors

Question

Let $P_k$ define an orthogonal projection on a ball of radius of $k$, and $x_i$, $y_i$ are two vectors. Now, I would like to bound: \begin{equation} ||\sum_{i=1}^n P_k(x_i) - P_k(y_i)||^2 \end{equation}

I know that I can bound it using Cauchy-schwarz inequality and VI such that: \begin{equation} ||\sum_{i=1}^n P_k(x_i) - P_k(y_k)||^2 \leq n\sum_{i=1}^n||P_k(x_i) - P_k(y_i)||^2 \leq n\sum_{i=1}^n ||x_i - y_i||^2 \end{equation}

However, I wonder if I can get a bound, e.g., $||\sum_{i=1}^n x_i-y_i||^2$.