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MiS Preprint

The set of gradients of a bump

Jan Kolár and Jan Kristensen


This is the first in a series of two papers.

The range of the gradient of a differentiable real-valued function with a non-empty and bounded support (i.e., a bump) is investigated. For a smooth bump ${}^{} f$ on ${\mathbb R}^2$ it is shown that the interior $\mathop{\rm int} \nabla f( {\mathbb R}^2)$ is connected and dense in $\nabla f( {\mathbb R}^2 )$. A purely topological characterization of such gradient ranges is however impossible. We give an example of a compact set $K \subset {\mathbb R}^2$ that is homeomorphic to the closed unit disk, but such that no affine image of $K$ is the gradient range of a smooth bump on ${\mathbb R}^2$. For smooth bumps on ${\mathbb R}^n$ we show that the gradient range cannot be separated by a totally disconnected set. The proof relies on a Morse-Sard type result involving irreducible separators of ${\mathbb R}^n$. Proofs are carried out for a class of ${\mathcal C}^1$ functions containing all those whose first order derivatives are Lipschitz or of bounded variation.

Finally, we present an example of a ${\mathcal C}^1$-smooth bump on $\ell_2$, which has a gradient range with non-empty and disconnected interior, and a ${\mathcal C}^{\infty}$-smooth weak bump on $\ell_2$ with the same property.

MSC Codes:
26B05, 26B30, 46G05, 46B20
gradient, range of derivative, bump, morse-sard theorem, critical set

Related publications

2002 Repository Open Access
Jan Kolar and Jan Kristensen

The set of gradients of a bump