(Almost isometric) local retracts in metric spaces (Q6611765)

The paper introduces the notions of a local retract, resp. almost isometric local retract (ai-local retract) as non-linear metric versions of ideals, resp. almost isometric ideals in Banach spaces.\N\NA subset \(N\subseteq M\) of a metric space is a \textit{local retract} if for every finite subset \(E\subseteq M\) and every \(\varepsilon>0\) there is a \((1+\varepsilon)\)-Lipschitz map \(f:E\to M\) satisfying \(f(e)=e\) for every \(e\in E\cap N\). \(N\) is an \textit{ai-almost retract} if we require the above map \(f\) to moreover satisfy \((1-\varepsilon)d(x,y)\leq d(f(x),f(y))\leq (1+\varepsilon)d(x,y)\) for every \(x,y\in E\).\N\NThe usefulness of these notions is demonstrated by the following results: If \(N\subseteq M\) is a local retract in a metric space and \(X\) is a proper metric space, then every Lipschitz function \(F:N\to M\) can be extended to a Lipschitz function \(\hat f:M\to X\) with the same Lipschitz constant. If \(X\) is moreoever a finite-dimensional Banach space, then this extension is a linear operator from \(\mathrm{Lip}_0(N,X)\) to \(\mathrm{Lip}_0(M,X)\), which are Banach spaces of Lipschitz functions preserving some distinguished point in \(N\). Analogously, when \(N\subseteq M\) is an ai-local retract, then isometric embeddings of \(N\) into proper metric spaces can be extended to isometric embeddings of \(M\). As an application, the authors obtain that for a local retract \(N\subseteq M\) in a metric space the Lipschitz-free space \(\mathcal{F}(N)\) is an ideal inside the Lipschitz-free space \(\mathcal{F}(M)\). Moreover, if \(M\) is proper, then \(N\) must be a \(1\)-Lipschitz retract in \(M\).\N\NThe paper moreover studies which metric spaces are absolute local retracts, resp. absolute ai-almost retracts, i.e., local retracts, resp. ai-local retracts in every metric space that contains them isometrically. Finally, it is shown that for every metric space \(M\) and a subset \(S\subseteq M\), there exists an ai-local retract \(S\subseteq N\subseteq M\) with the same density character as \(S\).