A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

2025/07/11 by Tu, Qiang, Mu, Xiaohuan, Guo, Tiexin +1 · 1 citation
#46A50 #46H25 #47H10 #60H25 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.08521

Abstract

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of L⁰-simplicial subdivisions of L⁰-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

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Cited by

• The random Kakutani fixed point theorem in random normed modules

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• Brouwer Fixed Point Theorem in (L⁰)^d