Suppose that ${\bf x}$, ${\bf y}$ are two vector variables, and ${\bf a}$ is a vector scalar, all of the same length. I wish to find $C$ such that: $$\|{\bf x} - {\bf a}\|_{2}^{2} - \|{\bf y} - {\bf a}\|_{2}^{2} \leq C\|{\bf x} - {\bf y}\|_{2}^{2}.$$
Note that it is related to the triangle inequality: $\|{\bf x} - {\bf a}\|_{2} - \|{\bf y} - {\bf a}\|_{2} \leq \left|\|{\bf x} - {\bf a}\|_{2} - \|{\bf y} - {\bf a}\|_{2}\right| \leq \|{\bf x} - {\bf y}\|_{2}$. However, if we have square operation, it seems difficult to find such a $C$.