In 1934, Whitney raised the question of how one can decide whether a function \(f\) defined on a closed subset \(X\) of \(\mathbb R^n\) is the restriction of a \(C^m\) function on \(\mathbb R^n\). He gave a characterization in dimension \(n=1\). The problem was fully solved by Fefferman in 2006.
In this talk, I will discuss a related conjecture: if a semialgebraic function \(f : X \to \mathbb R\) has a \(C^m\) extension to \(\mathbb R^n\), then it has a semialgebraic \(C^m\) extension. In particular, I will show that the \(C^{1,\omega}\) case of the conjecture is true, even in o-minimal expansions of the real field, where \(\omega\) is a definable modulus of continuity.
The proof is based on definable Lipschitz selections for affine-set valued maps.
This is joint work with Adam Parusinski.