Minimal Extension for the $\alpha$-Manhattan norm

TL;DR

This paper constructs a piecewise affine homeomorphism that closely approximates minimal directional derivatives in a convex polygon, extending previous results to more general domain shapes.

Contribution

It introduces a method to approximate minimal directional derivatives with finitely piecewise affine homeomorphisms on convex polygons, generalizing prior shape-specific results.

Findings

Achieves close approximation of minimal directional derivatives.

Extends previous shape-specific results to general convex polygons.

Provides a constructive method for such homeomorphisms.

Abstract

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.