Consider the linear programming problem \begin{align} f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1 \end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ is a $c \times 1$vector. Here $x=[x_1,\dots,x_n]$ is the $n \times 1$ vector to be found. Thus, in addition to the box constraint $0\leq x_i\leq 1$, we have $c$ constraints. It is also known that $p,A,b$ are all element-wise positive.

I randomly generated $p,A,b$ such that each entry is i.i.d and is uniformly distributed in the interval $[0,1]$. Note that this meets the problem set-up. I observed that whenever the problem was feasible, the optimal solution $x^*=[x_1^*,\dots,x_n^*]$ had a interesting property. The number of fractional $x_i^*$ (i.e. $0<x_i^*<1$) were atmost $c$. Every other $x_i^*$ belonged to $\{0,1\}$ (accounting for rounding). Is there anything going on here?