Pacific Northwest Probability Seminar: Gravitational Allocation to Uniform Points on the Sphere

Опубликовано 20 ноября 2017, 4:38

Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition - with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at ams.org/publications/journals/... Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos, and Tusnady (Combinatorica 1984). Joint work with Nina Holden and Alex Zhai. 

See more on this video at microsoft.com/en-us/research/v...