Is the direction of the longest line of a polytope unique? The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. 
The affine subspace is given by:
$X  \mbox{ u} = y$
where 
$u$ ∈ $\mathbb{R}^p$.
$X$ is a $m$ × $p$ matrix with $m$ < $p$ and 
$y$ is a $m$ dimensional vector.
The hypercube is given by:
$0$ ≤ $u$ ≤ $u_{max}$ 
The longest line of this polytope is clearly one of the lines that join the vertices of the polytope. My question is:
Is the direction of the longest line independent of $u_{max} \mbox{&nbsp}$ for some $u_{max}$ ≥ u* ?
Some simulations I did in Matlab indicates that the answer to the above question is yes but I am not sure if this will hold in general. I am assuming of course that the polytope actually resides in the hypercube defined by $0$ ≤ $u$ ≤ $u_{max}$. 
Any pointers to relevant literature in applied math or some approaches to answer the question would be very helpful.
Thanks
 A: I think the answer to your question is yes.  edit: NO
First I'll set some notation.  Assume that $X$ is rank $m$.  I'll denote by $L$ the m-dimensional plane defined by $X u = y$.  Subscripts will denote components of vectors.  Instead of $u_{max}$ I'll use $v$.  I'll denote by $A_{v}$ the hypercube $0\leq u_i\leq v$ for $1\leq i\leq p$.  The problem as stated is about the intersection between $A_v$ and $L$, which is a polytope I'll call $P_v$.  
We can rescale the coordinates by taking $u\rightarrow vu'$ so that $A_v$ has side length 1 in the $u'$ coordinates.  Under this transformation, $L$ keeps its orientation but is shifted.  In particular, $L$ is now defined by $X (vu')=y$, or $X u'=y'$ where $y' = 1/v*y$.  As $v$ gets larger and larger, $y'$ gets closer and closer to the origin.  Note that if $y$ were the zero vector, your problem is scale invariant and hence has a positive answer.  
If $y$ was not the zero vector, then to understand what $P_v$ looks like for large $v$, we need to understand how a slice through the hypercube behaves very close to one of its vertices.  Is there a result (for convex polytopes in general?) that tells us that the "shape" of a slice is stable to small translations of the slicing plane when we're close to a vertex?  I haven't found any counter-examples in the low-dimensional cases I've (unsystematically) tried.
edit: I spoke way to soon.  Consider a plane slicing through the 3-dimensional cube such that the plane makes right angles with the top and bottom faces of the cube.  In general the intersection will be a rectangle whose aspect ratio changes and becomes skinnier and skinnier as the plane gets closer to a vertex.  The direction of the longest segment in this rectangle (either of the diagonals) obviously does not stabilize.  But is there a positive result lurking here for suitably "generic" planes?
