Another note on the gradient problem of C. E. Weil

Abstract

Assume that \(G\subset \mathbb{R}^n\) is open and \(f:G\to\mathbb{R}\) is a differentiable function. C. E. Weil raised the gradient problem. In this problem it is asked whether \(\triangledown f\) satisfies the natural multidimensional generalization of the Denjoy-Clarkson property. We verify that if there are two dimensional counterexample functions to the gradient problem then their range should satisfy certain paradoxical convexity properties and the inverse image of ``many" values of \(\triangledown f\) is of positive linear measure.