Examples of absolutely minimal polyhedral chains in maximum norm spaces

We construct explicit area-minimizing polyhedral disks in ${\mathbb{R}}^n$ with the maximum norm for $n=2,3,4$. Their boundaries are polygonal images of the geodesic circle of length $2\pi$, obtained from $n$ distance functions. Piecewise constant calibrations show that these disks minimize mass among Ambrosio–Kirchheim metric currents with the same boundary. The example for $n=4$ gives the lower bound $11{\pi }^2\mathrm{/}20\approx 5.42828$ for Gromov’s filling area conjecture. For $n>4$, finite linear programs provide further lower bounds.