The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces
Author:
Pavel Shvartsman
Journal:
Trans. Amer. Math. Soc. 360 (2008), 5529-5550
MSC (2000):
Primary 46E35; Secondary 52A35, 54C60, 54C65
DOI:
https://doi.org/10.1090/S0002-9947-08-04469-3
Published electronically:
April 9, 2008
MathSciNet review:
2415084
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Abstract | References | Similar Articles | Additional Information
Abstract: We study a variant of the Whitney extension problem (1934) for the space . We identify
with a space of Lipschitz mappings from
into the space
of polynomial fields on
equipped with a certain metric. This identification allows us to reformulate the Whitney problem for
as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of
. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for
due to C. Fefferman.
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Additional Information
Pavel Shvartsman
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email:
pshv@tx.technion.ac.il
DOI:
https://doi.org/10.1090/S0002-9947-08-04469-3
Keywords:
Whitney's extension problem,
smooth functions,
finiteness,
metric,
jet-space,
set-valued mapping,
Lipschitz selection
Received by editor(s):
March 20, 2006
Received by editor(s) in revised form:
November 29, 2006
Published electronically:
April 9, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.