An optimal Brouwer's fixed point theorem for discontinuous functions

2025/12/16 by Adams, Henry, Frick, Florian
Computer Science · Mathematics · #FOS: Mathematics #Fixed Point Theorems Analysis #Mathematics and Applications #Metric Geometry (math.MG) #Optimization and Variational Analysis

paper · doi:10.48550/arxiv.2512.14934

openalex publication_date 2025/12/16 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28

Abstract

Brouwer's fixed point theorem states that any continuous function from a closed n-dimensional ball to itself has a fixed point. In 1961, Klee showed that if such a function has discontinuities that are bounded, then it has a point that is close to being fixed. We improve upon Klee's results in any finite-dimensional Euclidean space, and prove that our bounds are the best possible.

Citations

• Quantifying discontinuity
• Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
• Using Brouwer's fixed point theorem
• Borsuk-Ulam type theorems for metric spaces

Related

• Some extensions of the Brouwer fixed point theorem
• An easily verifiable proof of the Brouwer fixed point theorem
• Using Brouwer's fixed point theorem
• Brouwer's fixed point theorem with sequentially at most one fixed point
• Constructive proof of Brouwer's fixed point theorem for sequentially locally non-constant functions