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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)

##### MSC:
 26B05 Continuity and differentiation questions