Prove or Disprove the following statements either true or false using mathematical equation.

Let N,p,kN. Let data points x1,,xNRp and means m1,,mkRp. We want to cluster the data points using the k-means algorithm with m1,,mk as initial cluster means. In general (i.e., no matter with which input data we start the algorithm), it holds that... The output of the k-means-algorithm does not depend on the initial cluster means, i.e., it is always the same for all initial choices of the cluster means. The k-means-algorithm returns a global minimum of the cluster energy function, if k2. Regardless of how data points are initially assigned to clusters, the outcome of the k-means-algorithm delivers the same cluster means, but the clusters may differ. The k-means-algorithm returns a clustering with exactly k nonempty clusters, if and only if in every iteration at least one data point is assigned to each cluster