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