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Revista Matemática Iberoamericana

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Volume 35, Issue 4, 2019, pp. 1027–1052
DOI: 10.4171/rmi/1077

Published online: 2019-05-16

Whitney’s extension problem in o-minimal structures

Matthias Aschenbrenner[1] and Athipat Thamrongthanyalak[2]

(1) University of California Los Angeles, USA
(2) Chulalongkorn University, Bangkok, Thailand

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of $\mathbb{R}^n$ is the restriction of a $C^m$-function on $\mathbb{R}^n$. A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the $C^1$-case of Whitney's extension problem in this context. Our main tool is a definable version of Michael's selection theorem, and we include other another application of this theorem, to solving linear equations in the ring of definable continuous functions.

Keywords: O-minimal structures, Whitney’s extension problem, Michael’s selection theorem

Aschenbrenner Matthias, Thamrongthanyalak Athipat: Whitney’s extension problem in o-minimal structures. Rev. Mat. Iberoam. 35 (2019), 1027-1052. doi: 10.4171/rmi/1077