[Paper Review] Constructive proof of Brouwer's fixed point theorem for sequentially locally non-constant functions

Yasuhito Tanaka|arXiv (Cornell University)|Mar 9, 2011

Mathematical and Theoretical Analysis3 references3 citations

TL;DR

This paper presents a constructive proof of Brouwer’s fixed point theorem for uniformly continuous, sequentially locally non-constant functions on an n-dimensional simplex, using the existence of approximate fixed points. It establishes that such functions have exact fixed points without relying on the fan theorem, and further shows that this version of Brouwer’s theorem is logically equivalent to Sperner’s lemma.

ABSTRACT

We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions implies Sperner's lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner's lemma, our Brouwer's fixed point theorem is equivalent to Sperner's lemma.

Motivation & Objective

• To provide a constructive proof of Brouwer’s fixed point theorem under a stronger condition than local non-constancy, namely sequential local non-constancy.
• To demonstrate that the existence of approximate fixed points, combined with sequential local non-constancy, implies the existence of an exact fixed point.
• To show that Brouwer’s fixed point theorem for uniformly continuous, sequentially locally non-constant functions implies Sperner’s lemma for a simplex.
• To establish logical equivalence between Brouwer’s fixed point theorem under these conditions and Sperner’s lemma, avoiding non-constructive principles like the fan theorem.

Proposed method

• Define sequential local non-constancy as a uniform condition on totally bounded sets: if two sequences in a set have their images under f approaching the identity, then the sequences themselves must converge to each other.
• Use the fact that uniformly continuous functions on a compact simplex admit approximate fixed points, i.e., for every ε > 0, there exists x such that |x − f(x)| < ε.
• Construct a function f from a triangulated simplex using Sperner labeling, where labels determine component-wise shifts by a small τ > 0.
• Define f on the simplex via convex combinations of vertex mappings: f(x) = ∑λ_i f(x^i), ensuring uniform continuity.
• Prove that this constructed f is sequentially locally non-constant by analyzing convergence of sequences with |f(z_m) − z_m| → 0.
• Show that a fixed point of f must lie in a fully labeled simplex, thereby deriving Sperner’s lemma from the fixed point existence.

Experimental results

Research questions

• RQ1 Can Brouwer’s fixed point theorem be constructively proved for uniformly continuous functions that are sequentially locally non-constant?
• RQ2 Does the existence of approximate fixed points, combined with sequential local non-constancy, guarantee an exact fixed point without invoking the fan theorem?
• RQ3 Is Brouwer’s fixed point theorem for sequentially locally non-constant functions logically equivalent to Sperner’s lemma?
• RQ4 Can a constructive proof of Brouwer’s theorem be achieved without relying on non-constructive principles like the fan theorem?
• RQ5 What is the relationship between sequential local non-constancy and other conditions such as local non-constancy or at most one fixed point?

Key findings

• A uniformly continuous, sequentially locally non-constant function f from an n-dimensional simplex into itself has a fixed point if it admits approximate fixed points.
• The condition of sequential local non-constancy is stronger than local non-constancy and ensures that sequences with vanishing |f(x_n) − x_n| must converge to each other.
• The constructed function f based on Sperner labeling is uniformly continuous and sequentially locally non-constant, and its fixed point implies the existence of a fully labeled simplex.
• The existence of a fixed point for such functions implies Sperner’s lemma, establishing a logical equivalence between the two principles.
• The proof avoids the fan theorem and relies only on constructive principles, making it valid in Bishop-style constructive mathematics.
• The result shows that Brouwer’s fixed point theorem for this class of functions is equivalent to Sperner’s lemma, both constructively and logically.