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.
Affine image of the convex hull is its subset
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linear-algebra
convex-analysis
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0In 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. – 2012-03-29