multivariate Cauchy-distribution

Would the expression \(p(x_i) \propto \left(1+x_i C_{ij} x_j\right)^{-\frac{n+1}{2}}\) with a parameter matrix \(C_{ij}\) be a sensible generalisation of the Cauchy-distribution for the multivariate case with \(n\) dimensions? Is there a generalisation of the Cauchy-Schwarz-inequality?