In this work we address the Ev-SVM model proposed by Pérez-Cruz et al. as an extension of the traditional v support vector classification model (v-SVM). Through an enhancement of the range of admissible values for the regularization parameter v, the Ev-SVM has been shown to be able to produce a wider variety of decision functions, giving rise to a better adaptability to the data. However, while a clear and intuitive geometric interpretation can be given for the v-SVM model as a nearest-point problem in reduced convex hulls (RCH-NPP), no previous work has been made in developing such intuition for the Ev-SVM model. In this paper we show how Ev-SVM can be reformulated as a geometrical problem that generalizes RCH-NPP, providing new insights into this model. Under this novel point of view, we propose the RapMinos algorithm, able to solve Ev-SVM more efficiently than the current methods. Furthermore, we show how RapMinos is able to address the Ev-SVM model for any choice of regularization norm lp ≥1 seamlessly, which further extends the SVM model flexibility beyond the usual Ev-SVM models.