Optimal Coffee shops, Numerical Integration and Kantorovich-Rubinstein duality

Suppose you want to open up 7 coffee shops so that people in the downtown area have to walk the least amount to get their morning coffee. That’s a classical problem in Optimal Transport, minimizing the Wasserstein distance between the sum of 7 Dirac measures and the (coffee-drinking) population density. But in reality things are trickier. If the 7 coffee shops go well, you want to open an 8th and a 9th and you want to remain optimal in this respect (and the first 7 are already fixed). We find optimal rates for this problem in ($W_2$) in all dimensions. Analytic Number Theory makes an appearance and, in fact, Optimal Transport can tell us something new about $\sqrt{2}$ . All of this is also related to the question of approximating an integral by sampling in a number of points and a conjectured extension of the Kantorovich-Rubinstein duality regarding the $W_1$ distance and testing of two measures against Lipschitz functions.