mixture of multivariate Gaussians is elliptical? Let $\mathbb{R}^N$ be the ambient space and let $D_1$ and $D_2$ be different multivariate Gaussian distributions with the same mean (assume they are independent). Suppose further that $D_1$ is restricted to a proper subspace (for simplicity we can take a line) of $\mathbb{R}^N$ while $D_2$ covers the whole space. The variances are different.
Consider the mixture of these two distributions: $pD_1+(1-p)D_2$. 
First question: since each of these is an elliptical distribution, is the mixture of these elliptical?
I suspect the answer is yes, but I am having trouble showing why. If it is, then the natural second question is: how can I characterize the above mixture as an elliptical distribution in an explicit way?

Note: An elliptical distribution $f$ is such that $f(x)=C(g)g(x^T\Sigma^{-1}x)\text{det}(\Sigma)^{\frac12}$, where $\Sigma$ is a positive definite matrix in $\mathbb{R}^{N\times N}$ and $g:[0,\infty]\to[0\infty]$ is such that $\int_0^\infty g(x)x^{N-1}<\infty$ and $C(g)$ is a normalization parameter.

Any general references on elliptical distributions would be fantastic.

As JPi said, you'll get an ugly beast. The set of points with probability density greater than some threshold is a filled ellipsoid for an elliptical distribution, but if you consider the case where $D_1$ is the distribution of $X+\varepsilon Y$ and $D_2$ of $Y-\varepsilon X$ for $X,Y$ iid normal variables and $\varepsilon$ small, the set of points with probability density greater than some threshold will be a non-convex "crosshairs" set.
The stability of elliptical distributions under linear combination mentioned in JPi's paper refers to addition of random variables (that is, convolution of distribution functions), not linear combination of distribution functions (that is, mixture of distributions).
On the other hand, if the covariance matrices are scalar multiples of each other, you will definitely get an elliptical mixture. It's a necessary and sufficient condition.
