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An algorithm for approximating the metric projection onto a superelliptic disk

  • Valerian-Alin Fodor
  • Virgilius-Aurelian Minuta
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Abstract

We study conditions under which the metric projection onto a strictly convex body can be approximated by the projections onto polygonal approximations (in the Hausdorff-Pompeiu sense) of the strictly convex body itself. We provide a necessary condition by proving that the union of all the vertices of the polygonal approximations must be dense in the boundary of that set. To showcase our result, given a superellipse (Lamé curve) of order p>1, we prove that a sequence of polygons obtained by radially projecting the torsion subgroup of the unit circle onto the superellipse, converges to the superellipse, in the Hausdorff-Pompeiu metric. Moreover, this leads to a globally convergent new algorithm useful in approximating the metric projection onto a superelliptic disk (the convex hull of a superellipse).

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Cite

BibTeXRIS
BibTeX
@misc{fodor2026algorithm,
  title = {{An algorithm for approximating the metric projection onto a superelliptic disk}},
  author = {Valerian-Alin Fodor and Virgilius-Aurelian Minuta},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.24599},
  eprint = {2605.24599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24599},
}
RIS
TY  - UNPB
TI  - An algorithm for approximating the metric projection onto a superelliptic disk
AU  - Valerian-Alin Fodor
AU  - Virgilius-Aurelian Minuta
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2605.24599
UR  - https://arxiv.org/abs/2605.24599
AB  - We study conditions under which the metric projection onto a strictly convex body can be approximated by the projections onto polygonal approximations (in the Hausdorff-Pompeiu sense) of the strictly convex body itself. We provide a necessary condition by proving that the union of all the vertices of the polygonal approximations must be dense in the boundary of that set. To showcase our result, given a superellipse (Lamé curve) of order p>1, we prove that a sequence of polygons obtained by radially projecting the torsion subgroup of the unit circle onto the superellipse, converges to the superellipse, in the Hausdorff-Pompeiu metric. Moreover, this leads to a globally convergent new algorithm useful in approximating the metric projection onto a superelliptic disk (the convex hull of a superellipse).
ER  -