I have two related questions.
First, consider a maximal independent set of vectors $\{v_1,\cdots,v_k\}$ in $k$ dimensional space. The rows of a square matrix $A$ are from those vectors. The origin is the unique solution of the homogeneous system $Ax=0$. Now if I write another linear system of inequalities using $Ax \leq b^+, Ax \geq b^-$, where $b^+ \geq 0, b^- \leq 0$, and $b^+$ is not necessarily equal to $-b^-$. Can anyone get me the idea how to show the resulted polytope must be a parallelotope?
Second, we know that finding the farthest vertex in Euclidean distance from the origin on a polytope is NP-hard, since the number of vertices grows exponentially fast. If we know the polytope is a parallelotope, is there any algorithm to find the farthest vertex more efficiently?
Thank you.