In this paper a new algorithm for multi-class classification is presented. The algorithm, that is named Vertex Feature Classification (VFC), maps input data sets into an ad-hoc built space, called "simplex space" in order to perform geometric classification. More precisely, each class is first associated to a specific vertex of the polytope computed in the feature space. Successively, pattern classification is performed according to the geometric arrangement of patterns in a higher dimensional feature space. The experimental results, carried out on datasets of the UCI Machine Learning Repository, demonstrate the accuracy of the new algorithm is comparable with KNN, VDA and SVM, without or with a little training phase. An important aspect of this algorithm is its training time, which takes often a few milliseconds. Furthermore, the algorithm is robust and computationally efficient.