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 $u_{max} \mbox{ }$ 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

`$u_i\le v_i$`

for each $i$. The terminology is off here, though: Hyperplanes are unbounded (and of dimension one less than the containing space). We are looking at a polytope which is the intersection of an affine subspace with the standard hypercube in $\mathbb{R}^p$. – Harald Hanche-Olsen Jan 8 '10 at 18:45