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Another note on the gradient problem of C. E. Weil. (English) Zbl 0940.26011
Let \(G\subset \mathbb{R}^n\) be an open set and \(f: G\to \mathbb{R}\) a differentiable function. For \(H\subset \mathbb{R}^n\) we denote \(\Delta (H)=\{x:\nabla f(x)\in H\}\). The question raised by Weil is whether for \(H\) open, \(\Delta (H)\) is always of positive \(n\)-dimensional measure whenever nonempty. Up to now only partial results are known and one of them is the theorem proved in this article: Suppose that \(G\subset \mathbb{R}^2\) is an open set and \(f:G\to \mathbb{R}\) a differentiable function. Let \(H\subset \mathbb{R}^2\) be an open set such that \(\Delta (H)\) is nonempty and of zero \(n\)-dimensional measure. Then there is a ball \(B\) such that \(B\setminus \operatorname {cl}(\nabla f(G))\) is a nonempty convex open set, and for each \(\mathbf{p}\) from \(B\cap \text{int}(\operatorname {cl}(\nabla f(G))\), which is also a nonempty open set, the linear measure of \(\Delta (\{\mathbf{p}\})\) is strictly positive. The proof uses a technical one-dimensional lemma which can be applied also to an alternative proof of the classical Denjoy-Clarkson property.
Reviewer: Jan Malý (Praha)

26B05 Continuity and differentiation questions