The distance between two distributions

The gold distribution μ is fixed at the top of the circle; drag the ice-blue one, ν, round the circle, and set each one's width with its slider. The circle has radius 1. Connes measures the distance like this: among all functions f that are not too steep, with slope at most 1 in size, find one whose average over ν differs most from its average over μ; that largest difference is the distance. The coral line is such an f, drawn once round inside the circle and again on the right, where the circle is cut open and laid flat; there it is made of straight pieces with slope +1 or −1. The corner works the distance out both ways: on the left the difference, on the right the least cost of moving μ onto ν, amount moved times distance moved round the circle, added up. The two numbers always agree. Make both distributions narrow and the distance approaches the shorter arc between their centres; drag ν to the bottom of the circle and that arc is π. With different widths, the distance is not 0 even when the centres coincide.