Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
64/2002

The set of gradients of a bump

Jan Kolár and Jan Kristensen

Abstract

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.

Received:
Aug 2, 2002
Published:
Aug 2, 2002
MSC Codes:
26B05, 26B30, 46G05, 46B20
Keywords:
gradient, range of derivative, bump, morse-sard theorem, critical set

Related publications

Preprint
2002 Repository Open Access
Jan Kolar and Jan Kristensen

The set of gradients of a bump