As the need for interpretable machine learning continues to grow, we propose a novel post-clustering method to generate counterfactual explanations for clustering results. Specifically, our method answers the question: What is the smallest change to a data point that would make it the medoid of its cluster? These explanations offer valuable insights in domains like healthcare and marketing, where identifying minimal adjustments to align individuals or customers with desirable cluster representative profiles can inform personalized interventions and strategic decision-making. By formulating the problem as a convex optimization model, specifically a second-order cone program, our method guarantees global optimality requiring readily available and effective solvers for practical implementation.