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: 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.