The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables

Zelený, Miroslav

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 405-428.

On construit une fonction différentiable $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) telle que l’ensemble ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ est non vide et sa dimension de Hausdorff est $1$. C’est une réponse à une question posée par Z. Buczolich.

We construct a differentiable function $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) such that the set ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.

Classification : 26B05, 28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game
Mot clés : propriété de Denjoy-Clarkson, gradient, mesure de Hausdorff, jeu infini

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