Optimal Coffee shops, Numerical Integration and Kantorovich-Rubinstein duality

Abstract

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.

Date
2021-01-30 10:00 AM PST
Speaker
Stefan Steinerberger (University of Washington (Math))
Location
Online (zoom)
Pacific Institute for the Mathematical Sciences

This event is part of the Pacific Interdisciplinary Hub on Optimal Transport (PIHOT) which is a collaborative research group (CRG) of the Pacific Institute for the Mathematical Sciences (PIMS).