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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.

Rodrigo de Azevedo
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entropy
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  • The intersection is bounded, so cannot b a hyperplane. It seems in 3-space a nontrivial intersection would be a circle. – coffeemath May 28 '23 at 10:47
  • @coffeemath I agree but what if I put the constraint that c_1 <c_2 in a lexicographic sense because the latter are vectors. – entropy May 28 '23 at 10:53
  • samir: That will not affect the type of intersection. It will affect the hyperplane which contains the intersection, since the orientation of the hyperplane is a bit restricted by $c_1<c_2$ assumption. – coffeemath May 28 '23 at 11:21

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