A bounded compactness theorem for \(L^ 1\)-embeddability of metric spaces in the plane (Q1196200)

A metric space \((W,d)\) is said to be \(L^ q\)-embeddable in \(\mathbb{R}^ k\) if there is a distance preserving map from \((W,d)\) into \((\mathbb{R}^ k,L^ q)\). A classical result of K. Menger says that \((W,d)\) is \(L^ 2\)- embeddable in \(\mathbb{R}^ k\) iff every \((k+3)\)-point subspace is \(L^ 2\)- embeddable in \(\mathbb{R}^ k\). It is natural to ask for analogs for other \(L^ q\)-metrics on \(\mathbb{R}^ k\). The authors solve a special case by showing that \((W,d)\) is \(L^ 1\)-embeddable in \(\mathbb{R}^ 2\) iff every 11- point subspace is so, conjecturing that 11 can be replaced by 6 here. Generalization to \(L^ 1\)-embeddability in \(\mathbb{R}^ k\) for \(k\geq 3\) is still an open problem. The authors discuss some problems in that direction and formulate two conjectures.