An example related to Whitney extension with almost minimal \(C^m\) norm (Q734609)

The first author [Rev. Mat. Iberoam. 25, No. 1, 275--421 (2009; Zbl 1172.65010)] has conjectured the following property: Fix \(m,n\) and a \(C^m\)-norm. Let \(E\subset\mathbb{R}^n\) be a finite set, \(f:E\rightarrow\mathbb{R}\) and \(M,\varepsilon>0\). Assume that for any subset S of E with \(|S|\leq{k}(\varepsilon)\) there exists a \(C^m\)-function \(F_{S}:\mathbb{R}^n\rightarrow\mathbb{R}\) such that the \(C^m\)-norm of \(F_S\) is bounded by \(M\) and \(F_{S}(x)=f(x)\) for all \(x\in S\). Then, there exists a \(C^m\)-function \(F:\mathbb{R}^n\rightarrow\mathbb{R}\) such that the \(C^m\)-norm of \(F\) is bounded by \((1+\varepsilon)M\) and \(F(x)=f(x)\) for all \(x\in E\). The constant \(k\) depends only on \(m,n,\varepsilon\) and on the choice of the \(C^m\)-norm. In this paper the authors present a counter-example in the case \(m=n=2\) showing that the conjecture is false.