Given $a ,b \in \Bbb R^+$ and ${\bf x}, {\bf y} \in \Bbb R^n$, where ${\bf x} \neq {\bf y}$, how can I characterize the following set? Is it a hyperplane?
$$ \left\{ {\bf c}_1 \in \Bbb R^n : \| {\bf x} - {\bf c}_1 \|_2 = a \right\} \cap \left\{ {\bf c}_2 \in \Bbb R^n : \| {\bf y} - {\bf c}_2 \|_2 = b \right\} $$
I am trying to solve $k$-means problem in an alternative way. Particularly, I want to show that assignment vector elements should not change too much in the Euclidean distance sense around the global optimum.