Colorful Borsuk--Ulam Theorems (Zoe Wellner, CMU)

Combinatorics Seminar
Friday, March 15, 2024 - 3:15pm for 1 hour (actually 50 minutes)
Zoe Wellner – Carnegie Mellon University – zwellner@andrew.cmu.edu
Evelyne Smith-Roberge

 The classical Borsuk--Ulam theorem states that for any continuous map  from the sphere to Euclidean space, $f\colon S^d\to R^d$, there is a pair of antipodal points that are identified, so $f(x)=f(-x)$. We prove a colorful generalization of the Borsuk–Ulam theorem. The classical result has many applications and consequences for combinatorics and discrete geometry and we in turn prove colorful generalizations of these consequences such as the colorful ham sandwich theorem, which allows us to prove a recent result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case, and Brouwer's fixed point theorem, which allows us to prove an alternative between KKM-covering results and Radon partition results.  This is joint work with Florian Frick.