Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.