Nearest-centroid decision hyperplane ($2(z_1-z_2{)}^{\mathsf{T}}x+\Vert z_2{\Vert }^2-\Vert z_1{\Vert }^2=0$)

For two distinct centroids in a Euclidean space, equal squared Euclidean distances expand to the displayed affine hyperplane. It is their perpendicular bisector; each closed half-space consists of points at least as close to the centroid on that side. This proves weak linear separability after a two-centroid nearest-assignment step, allowing ties on the boundary. Coincident centroids are an essential exception: every point ties and the distance comparison supplies no nonzero normal vector.