Let X be a metric space with a base point 0, and let \(\textrm{Lip}_0(X)\) be the Banach space of all Lipschitz functions \(f:X\rightarrow \mathbb {R}\) such that \(f(0)=0\) . Given a set of points \(\left( (x_i,y_i)\right) _{i\in I}\) in \(X^2\) with \(x_i\ne y_i\) for all \(i\in I\) , we study the following interpolation problem: when for each bounded set \(\left( \alpha _i\right) _{i\in I}\) in \(\mathbb {R}\) the algorithm \( \frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\alpha _i\qquad (i\in I) \) can be implemented by a function \(f\in \textrm{Lip}_0(X)\) ? Our approach involves the concept of a Beurling set of functions in \(\textrm{Lip}_0(X)\) for \(\left( (x_i,y_i)\right) _{i\in I}\) which has shown to be useful in the so-called transportation problem.
