Minimal extension for the $\alpha$-Manhattan norm

Let partial derivative Q be the boundary of a convex polygon in R-2, e(alpha) = (cos alpha, sin alpha) and e(alpha)(perpendicular to) = (- sin alpha, cos alpha) a basis of R-2 for some alpha is an element of [0, 2 pi) and phi : partial derivative Q -> R-2 a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism v : Q -> R-2 coinciding with phi on partial derivative Q such that the following property holds: l(D-u, e(alpha))l(Q) (resp., (Du, e(alpha)(perpendicular to))l(Q)) is as close as we want to inf l(D-u, e(alpha))l(Q) (resp., inf l(Du, e(alpha)(perpendicular to))l(Q)) where the infimum is meant over the class of all BV homeomorphisms u extending phi inside Q. This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511-555] in the shape of the domain.