I am dealing with the following standard linear program
\begin{align} & \max c^T x \\[6pt] & \text{s.t. } Ax=b \\ & lb<x<ub \end{align}
The problem's matrix has high dimensionality and is under-determined, in a way that several $x$ solutions can satisfy the optimal maximal objective (Alternative optimal solutions).
Now my question surrounds the distribution of the $x$ variables. Do they follow a normal distribution in the $[lb,ub]$ interval, provided I obtain a large number of solutions?
My first guess would be positive if we consider the central limit theory, but I am not sure.
The set of optimal solutions is a polyhedron, probably with much lower dimension than the number of variables. You didn't tell us anything about this except that it is not a single point. It may still be a line segment, for example.
One important question may be how those solutions are obtained. For example, if you're using some version of the simplex method, the solutions you obtain will always be basic solutions (thus extreme points of the set of optimal solutions).
But in any case, I see no reason to believe that the results should follow a normal distribution.
EDIT: It may also be worth mentioning that even if the matrix is under-determined, it is typical to have a unique optimal solution. The feasible region has a finite number of extreme points, and if you took a random $c$ the probability that two extreme points have the same value of $c^T x$ would be $0$.