An affine map $(t,M)$ with $t\in R^3$ and matrix $M$ maps $x\in R^3$ into $t+Mx$. It has property $P$ if for any $x$ with $|x|\leq 1$ then $|t+Mx|\leq 1$.
Our goal is to characterize the set of such maps with property $P$, that is characterized the set of $(t,M)$.
My question here is to give polynomial constraint of the above property $P$ if it exists.
Polar Decomposition on M, M=UJ, where U is unity and J is positive. Suppose t=U s.
Then |t+Mx| =|s+Jx|=~)
U can do this:)