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)*)

Event

Location

Online (zoom)

*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).*