An application of Brouwer's fixed-point theorem: continuously differentiable convex functions with gradient of constant norm
Csaba Vincze
TL;DR
The paper proves that a continuously differentiable convex function with gradient of constant norm on $\mathbb{R}^n$ must be affine, providing a first-order characterization that does not rely on second-order conditions. It achieves this by combining Brouwer's fixed-point theorem, the Cauchy–Schwarz inequality, and the first-order convexity inequality to show that the gradient remains constant along lines in the gradient direction, ultimately yielding $f(u)=\langle c_1,u\rangle+c_0$. The differentiability condition is shown to be essential, with distance-function-type examples illustrating potential singular behavior otherwise, and the argument is extended to Hilbert spaces via Browder–Minty theory. A Hilbert-space generalization is sketched, indicating the approach's broader applicability to convex, differentiable functions beyond finite-dimensional Euclidean spaces.
Abstract
As an application of Brouwer's fixed-point theorem we prove that a continuously differentiable convex function with gradient of constant norm is an affine mapping. It is a first-order characterization of affine mappings among continuously differentiable convex functions, because neither the second-order condition of convexity nor related operators are used. The condition of differentiability is essential as the case of the norm function shows. In addition to Brouwer's theorem, the proof is based on the Cauchy--Bunyakovsky--Schwarz inequality and becomes complete by minimizing the distance between lines of gradient directions. Following the steps of the proof, we sketch a possible generalization of the result to functions defined on Hilbert spaces.