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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 Rn 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 11π2/20≈5.42828 for Gromov's filling area conjecture. For n>4, finite linear programs provide further lower bounds.

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Cite

BibTeXRIS
BibTeX
@misc{zust2026examples,
  title = {{Examples of absolutely minimal polyhedral chains in maximum norm spaces}},
  author = {Roger Züst},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29593},
  eprint = {2610.29593},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29593},
}
RIS
TY  - UNPB
TI  - Examples of absolutely minimal polyhedral chains in maximum norm spaces
AU  - Roger Züst
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29593
UR  - https://arxiv.org/abs/2610.29593
AB  - 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≈5.42828 for Gromov's filling area conjecture. For n>4, finite linear programs provide further lower bounds.
ER  -