# Affine image of the convex hull is its subset

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.

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Well, $Av_k+b\in\mathcal P$ for all $k$ is a necessary and sufficient condition, but I'd surprised if you find anything general and useful beyond. If you like, you can rephrase that as $b\in\bigcap_k(\mathcal P-Av_k)$. – joriki Mar 29 '12 at 11:04
@joriki: thanks - I was also expecting some necessary conditions on $A$ - say that the spectral radius $\rho(A)\leq 1$ to avoid expansions – Ilya Mar 29 '12 at 17:19
In fact $|\det A|\le1$, since the volume is multiplied by $|\det A|$ and must fit into the original volume. Both of those bounds are obviously tight since $A$ can be the identity. – joriki Mar 29 '12 at 17:56