1. Suboptimality of k-Means: For every parameter t > 1, show that there exists an instance of the k-means problem for which the k-means algorithm (might) find a solution whose k-means objective is at least t · OPT, where OPT is the minimum k-means objective.

2 k-Means Might Not Necessarily Converge to a Local Minimum: Show that the kmeans algorithm might converge to a point which is not a local minimum. Hint: Suppose that k = 2 and the sample points are {1,2,3,4} ⊂ R suppose we initialize the k-means with the centers {2,4}; and suppose we break ties in the definition of Ci by assigning i to be the smallest value in argminj x−μj       .