Affine image of the convex hull is its subset

Question

For a set of points $v_1,v_2,\dots, v_r\in \mathbb R^n$ let's use $\mathcal P(v_1,\dots,v_r)$ to denote the convex hull of these points. Let us consider $A\in\mathbb R^{n\times n}$ and $b\in \mathbb R^n$ and denote $$ f(\mathcal P) = \{Ax+b:x\in \mathcal P\}. $$ Given $v_1,\dots, v_r$ I wonder about necessary and/or sufficient conditions on $A,b$ for $$ f(\mathcal P) \subset \mathcal P.\tag{1} $$ I am completely new to this area - just recently read the paper which considers the system $$ x_{k+1} = Ax_k+b $$ in $\mathcal P$ but there is no justification given on why should the dynamics always be inside $\mathcal P$. Of course, $(1)$ is a necessary and sufficient for that, but now I wonder how restrictive is condition $(1)$ on $A$ and $b$. Maybe the problem and solution are well-known - and please, feel free to retag.