  Although an input distribution may not majorize a target distribution, it may
majorize a distribution which is close to the target. Here we introduce a
notion of approximate majorization. For any distribution, and given a distance
$\delta$, we find the approximate distributions which majorize (are majorized
by) all other distributions within the distance $\delta$. We call these the
steepest and flattest approximation. This enables one to compute how close one
can get to a given target distribution under a process governed by
majorization. We show that the flattest and steepest approximations preserve
ordering under majorization. Furthermore, we give a notion of majorization
distance. This has applications ranging from thermodynamics, entanglement
theory, and economics.
