Colorful Borsuk--Ulam theorems and applications

We prove a colorful generalization of the Borsuk--Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, con...

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Bibliographic Details
Published in:arXiv.org 2023-09
Main Authors: Frick, Florian, Wellner, Zoe
Format: Article
Language:English
Subjects:
Fixed points (mathematics)
Theorems
Online Access:Get full text
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Summary:We prove a colorful generalization of the Borsuk--Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case. We prove a colorful generalization of Fan's antipodal sphere covering theorem, we derive a short proof of Gale's colorful KKM theorem, and we prove a colorful generalization of Brouwer's fixed point theorem. Our results also provide an alternative between Radon-type intersection results and KKM-type covering results. Finally, we prove colorful Borsuk--Ulam theorems for higher symmetry.
ISSN:2331-8422