Let the set $\mathcal{S}$ be the Minkowski Sum of the two following sets, i.e. a Zonohedron ($\mathcal{Z}\subseteq \mathbb{R}^3$) and ($\mathcal{P}_O\cap\mathcal{B}(0,r)$), with $\mathcal{P}_O\subseteq \mathbb{R}^3$ being a Closed Convex Polyhedral Cone, pointed in $0$, and $\mathcal{B}(0,\rho)\subseteq \mathbb{R}^3$ the closed ball of radius $\rho>0$ and centre $0$. Clearly, $S$ is convex and compact.
My goal is to find the boundary of the set $\mathcal{T}$, related by a linear mapping to $\mathcal{S}$, i.e. $\mathcal{T}=\{x\in\mathbb{R}^6\,|\,x=Ay, \forall y\in \mathcal{S}\}$. Both $\mathcal{S}$ and $A\in \mathbb{R}^{6\times3}$ are known in this problem. The matrix $A$ is given by: $A=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ 0&-r_z&r_y\\ r_z&0&-r_x\\ -r_y&r_x&0 \end{bmatrix}$, where $(r_x,r_y,r_z)$ are the components of a given vector.
Generally, $\partial \mathcal{T}\ne A(\partial \mathcal{S})$ since $A$ is rectangular. However, is there any special case, i.e. sufficient condition, where the latter inequality holds as equality even with $A$ being rectangular? If not, any advice about how I may simplify the problem would be strongly helpful.
Thanks in advance!