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The convex clustering formulation of Chi and Lange (2015) is revisited. While this formulation can be precisely and efficiently solved, it uses the standard Euclidean metric to measure the distance between the data points and their corresponding cluster centers and hence its performance deteriorates significantly in the presence of outlier features. To address this issue, this paper considers a formulation that combines convex clustering with metric learning. It is shown that: (1) for any given positive definite Mahalanobis distance metric, the problem of convex clustering can be precisely and efficiently solved using the Alternating Direction Method of Multipliers; (2) the problem of learning a positive definite Mahalanobis distance metric admits a closed-form solution; (3) an algorithm that alternates between convex clustering and metric learning can provide a significant performance boost over not only the original convex clustering formulation but also the recently proposed robust convex clustering formulation of Wang et al. (2017).