The geometric framework for the support vector machine (SVM) classification problem provides an intuitive ground for the understanding and the application of geometric optimization algorithms, leading to practical solutions of real world classification problems. In this work, the notion of "reduced convex hull" is employed and supported by a set of new theoretical results. These results allow existing geometric algorithms to be directly and practically applied to solve not only separable, but also nonseparable classification problems both accurately and efficiently. As a practical application of the new theoretical results, a known geometric algorithm has been employed and transformed accordingly to solve nonseparable problems successfully.