Gradient ranges of bumps on the plane

Authors:
Jan Kolár and Jan Kristensen

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1699-1706

MSC (2000):
Primary 26B05; Secondary 46G05

Published electronically:
December 20, 2004

MathSciNet review:
2120251

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Abstract | References | Similar Articles | Additional Information

Abstract: For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

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Additional Information

**Jan Kolár**

Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Email:
kolar@karlin.mff.cuni.cz

**Jan Kristensen**

Affiliation:
Mathematical Institute, 24-29 St Giles’, University of Oxford, Oxford OX1 3LB, United Kingdom

Email:
kristens@maths.ox.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-04-07747-0

Keywords:
Gradient range,
derivative,
bump,
Morse-Sard theorem

Received by editor(s):
November 5, 2002

Received by editor(s) in revised form:
February 2, 2004

Published electronically:
December 20, 2004

Communicated by:
David Preiss

Article copyright:
© Copyright 2004
American Mathematical Society