Wasserstein Distance: the Cheapest Way to Move the Earth

Think of p, on the left wall, as piles of earth, and q, on the back wall, as holes to fill. The question is how much work it takes to move one into the other.

A transport plan is a coupling: a joint distribution on the floor, whose rows add up to p and whose columns add up to q. The mass at a point x, y travels from x to y, so its cost is its distance from the diagonal.

The rest of the narration is in the book's Companion.