multivariate Cauchy-distribution

Would the expression $p(x_i)\propto {(1+x_i{C}_{ij}{x}_j)}^{-\frac{n+1}{2}}$ 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?