 global minimum.'";
                case 5:
                    return "The sequence of objective function values produced by the \\(k\\)-means algorithm is non-increasing.";
                case 6:
                    return "Note that we used the fact that the square root (and the square) is non-decreasing to conclude that minimizing \\(\\|\\mathbf{x}_j - \\boldsymbol{\\mu}_i\\|^2\\) or its square root \\(\\|\\mathbf{x}_j - \\boldsymbol{\\mu}_i\\|\\) are equivalent.";
                case 7:
             