Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references:

Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, \dots, x_m) \in \mathbb{R}^m : x_1 A_1 + \dots + x_m A_m \preceq B\},$$ where $x_1 A_1 + \dots + x_m A_m \preceq B$ means that $B - x_1 A_1 - \dots - x_m A_m$ is positive semi-definite. It can be proven that $S$ is a convex set. This can be done easily by defining an affine map on a suitable convex set in the space of $n\times n$ symmetric matrices so that the inverse image of this convex set is $S$.

Let $$\mathcal{P} = \{x\in\mathbb{R}^n : Ax \le b\}$$ be a rational convex polytope. A lot can be said about $L_\mathcal{P}(t) = \vert \{\mathbb{Z}^n \cap t\mathcal{P}\}\vert$. What is known about $$\vert \{(x_1, \dots, x_m) \in \mathbb{Z}^m: x_1 A_1 + \dots + x_m A_m \preceq tB\}\vert,$$ where $t\in\mathbb{N}$?

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Counting lattice points in this context would be counting lattice points in an "orbitope" (assuming boundedness): arxiv.org/pdf/0911.5436.pdf. –  Steven Collazos Dec 5 '12 at 21:15