Complexity of some problems of quadratic partitioning of a finite set of points in Euclidean space into balanced clusters

We consider three related problems of partitioning an $N$-element set of points in $d$-dimensional Euclidean space into two clusters balancing the value of the intracluster quadratic variance normalized by the cluster size in the first problem, the intracluster quadratic variance in the second problem, and the size-weighted intracluster variance in the third problem. The NP-completeness of all these problems is proved.