Kmeans on “symmetric” data

A set is said to be fully-symmetric if for every $x$ in it, negating one of its components results in $y$ such that $y$ is in the set as well.

A set is said to be semi-symmetric if for every $x$ in it, negating all of its components (at once) results in $y$ such that $y$ is in the set as well.

Now examine the optimal solution of the Kmeans objective with $K=2d+1$ for d-dimensional unique observations that are fully-symmetric.

Suppose it is known that the optimal means set w.r.t the above setup is unique and contains the zero vector. Prove or give a counter example to the following claim: The set of optimal means is semi-symmetric

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– cactus314 Feb 22 '13 at 20:29