A key procedure in proximal bundle methods for convex minimization problems is the definition of stability centers, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-center classification rule for proximal bundle methods. We show that the proposed bundle variant has three particularly interesting features: (i) the sequence of stability centers generated by the method converges strongly to the solution that lies closest to the initial point; (ii) the entire sequence of stability centers is contained in a ball with diameter equal to the distance between the initial point and the solution set; (iii) if the sequence of stability centers is finite, $\hat{x}$ being its last element, then the sequence of non-stability centers (null steps) converges strongly to $\hat{x}$. Property (i) is useful in some practical applications in which a minimal norm solution is requested. We show the interest of this property on several instances of a full sized unit-commitment problem.