I am interested in a family of multivariate distributions that can be seen as a generalization of the multivariate normal distribution, insofar as they are defined by an expectation value $\vec \mu$ and a covariance matrix $\Sigma$, plus a monotonously decreasing function $g(d)$ such that the density is $$ p(\vec x) \propto g \left ( \Delta(\vec x, \vec \mu) \right ) $$ where $$ \Delta(\vec a, \vec b) = \sqrt { (\vec a - \vec b)^T \Sigma^{-1} (\vec a - \vec b) } $$ is the Mahalanobis distance. The multivariate normal is of course recovered by $g(d) = \exp (- \frac12 d^2 )$.
My first question is: What is the name of this family of distributions?
It is simple to show that for classification of a given data point to one of two or more classes, each of which is described by such a density with different $\mu$ but identical $\Sigma$ and $g(d)$, optimal classification boundaries are piecewise linear (hyperplanar).
My second question is: Is this a standard result, and if yes, what is the standard literature (textbook) reference for it?