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PreprintCC BY 4.0arXiv

On Universal Minkowski Classes of Polytopes: Theory and Applications

  • Boris Houska
  • Matthias A. Müller
  • Mario Eduardo Villanueva
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Abstract

This paper establishes the existence, in every dimension, of universal Minkowski classes of configuration-constrained polytopes, with direct applications to set-theoretic control. For any prescribed accuracy, such a class contains an approximation of every compact convex set to that accuracy and admits a jointly affine parameterization of facets and vertices. The construction is explicit and relies on a combinatorial averaging procedure that produces minimal configuration enclosures. In control, these polytopic templates enable convex reformulations of otherwise non-convex problems, including invariant-set computation and control Lyapunov function design. Tutorial examples illustrate both the geometric construction and its control applications.

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Cite

BibTeXRIS
BibTeX
@misc{houska2026universal,
  title = {{On Universal Minkowski Classes of Polytopes: Theory and Applications}},
  author = {Boris Houska and Matthias A. Müller and Mario Eduardo Villanueva},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41966},
  eprint = {2610.41966},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41966},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Universal Minkowski Classes of Polytopes: Theory and Applications
AU  - Boris Houska
AU  - Matthias A. Müller
AU  - Mario Eduardo Villanueva
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41966
UR  - https://arxiv.org/abs/2610.41966
AB  - This paper establishes the existence, in every dimension, of universal Minkowski classes of configuration-constrained polytopes, with direct applications to set-theoretic control. For any prescribed accuracy, such a class contains an approximation of every compact convex set to that accuracy and admits a jointly affine parameterization of facets and vertices. The construction is explicit and relies on a combinatorial averaging procedure that produces minimal configuration enclosures. In control, these polytopic templates enable convex reformulations of otherwise non-convex problems, including invariant-set computation and control Lyapunov function design. Tutorial examples illustrate both the geometric construction and its control applications.
ER  -