Whitney’s extension problem in o-minimal structures

Abstract

In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of R n is the restriction of a C m-function on 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 applications of this theorem, to solving linear equations in the ring of definable continuous functions.

$C^m$ Semialgebraic Sections Over the Plane

“…Problem 1 in the setting of C 1 loc (R n ) was settled affirmatively by M. Aschenbrenner and A. Thamrongthanyalak [1]. Our results on Problem 3 imply an affirmative solution for…”

On p‐adic semi‐algebraic continuous selections

“…This paper discusses the above problem in the p ‐adic semi‐algebraic context. (Note that similar questions in the context of the reals were discussed in [2, 3, 7, 21]. )…”

“…It is easy to see that for each , . In o‐minimal expansions of the real field, we know that this sequence of iterated Glaeser refinements of set‐valued maps is eventually stable (cf., e.g., [2 19]). Here, we shall show that the same result also holds in the p ‐adic semi‐algebraic context.…”

“…This paper discusses the above problem in the p-adic semi-algebraic context. (Note that similar questions in the context of the reals were discussed in [2,3,7,21].) Throughout, let p be a fixed prime number, Q p be the set of p-adic numbers with the p-adic valuation v : Q p → Z ∪ {+∞} and E denote a subset of some Q n p .…”

Definable continuous selections of set-valued maps in o-minimal expansions of the real field

“…3.4 (M. Aschenbrenner, A. Thamrongthanyalak [1]). There is a cell decomposition C of E such that T C is lower semicontinuous for every C ∈ C.…”