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arXiv · 2610.29593

Examples of absolutely minimal polyhedral chains in maximum norm spaces

Roger Züst
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Abstract

We construct explicit area-minimizing polyhedral disks in $\R^n$ with the maximum norm for $n=2,3,4$. Their boundaries are polygonal images of the geodesic circle of length $2π$, 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 $11π^2/20\approx5.42828$ for Gromov's filling area conjecture. For $n>4$, finite linear programs provide further lower bounds.

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Roger Züst. 2026-10-21. Examples of absolutely minimal polyhedral chains in maximum norm spaces. https://arxiv.org/abs/2610.29593

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