  In this paper, we extend the methodology developed for Support Vector
Machines (SVM) using $\ell_2$-norm ($\ell_2$-SVM) to the more general case of
$\ell_p$-norms with $p\ge 1$ ($\ell_p$-SVM). The resulting primal and dual
problems are formulated as mathematical programming problems; namely, in the
primal case, as a second order cone optimization problem and in the dual case,
as a polynomial optimization problem involving homogeneous polynomials.
Scalability of the primal problem is obtained via general transformations based
on the expansion of functionals in Schauder spaces. The concept of Kernel
function, widely applied in $\ell_2$-SVM, is extended to the more general case
by defining a new operator called multidimensional Kernel. This object gives
rise to reformulations of dual problems, in a transformed space of the original
data, which are solved by a moment-sdp based approach. The results of some
computational experiments on real-world datasets are presented showing rather
good behavior in terms of standard indicators such a \textit{accuracy index}
and its ability to classify new data.
