Let $\partial \mathcal{ Q}$ be the boundary of a convex polygon in $\mathbb{ R}^2$, $e_\alpha = (\cos\alpha, \sin \alpha)$ and $e_{ \alpha}^{ \bot} = (-\sin\alpha , \cos \alpha)$ be a basis of $\mathbb{ R}^2$ for some $\alpha\in[0,2\pi)$ and $\phi:\partial\mathcal{ Q} \to\mathbb{ R}^2$ be a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism $v: \mathcal{ Q} \to \mathbb{ R}^2$ coinciding with $\phi$ on $\partial \mathcal{ Q}$ such that the following property holds: $|\langle Dv, e_{ \alpha}\rangle|(\mathcal{ Q})$ (resp. $\langle Dv, e_{ \alpha}^{ \bot}\rangle|(\mathcal{ Q})$) is as close as we want to $\inf |\langle Du, e_{ \alpha}\rangle|(\mathcal{ Q})$ (resp. $\inf |\langle Du, e_{ \alpha}^{ \bot}\rangle|(\mathcal{ Q})$) where the infimum is meant over the class of all $BV$ homeomorphisms $u$ extending $\phi$ inside $\mathcal{ Q}$. This result extends that already proven in [14] in the shape of the domain.
Categories: math.AP