Colorful Borsuk–Ulam Theorems

The classical Borsuk–Ulam theorem states that for any continuous map from the sphere to Euclidean space, f : S^d → 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 Bárány, Hubard, and Jerónimo on well-separated measures as a special case, and extend that case to the chessboard partition setting. The consequences on Brouwer’s fixed point theorem additionally allow us to prove an alternative between KKM- covering results and Radon partition results.