Absolute Lipschitz extendability and linear projection constants

We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants of certain Lipschitz-free spaces. As an application, we show that $\mbox{ae}(3)=4/3$ and $\mbox{ae}(4)\geq (5+4\sqrt{2})/7$. Moreover, we discuss how to compute relative projection constants by solving linear programming problems.