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PreprintCC BY 4.0arXiv

Facial distance and diameter

  • Antoine Deza
  • David Martínez-Rubio
  • Javier Peña
  • Elias Wirth
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Abstract

The facial distance of a polytope, equivalently its pyramidal width, is a geometric condition measure that appears in the convergence bounds for Frank--Wolfe algorithms. Peña and Wirth developed calculus rules for this quantity and conjectured that the ratio of the facial distance to the diameter multiplied by the square root of the dimension is bounded by a constant. We show that this quantity is uniquely maximized by a regular simplex in each dimension, which implies that the conjecture holds and that √2 is the smallest constant. We also compute the facial distance of the Birkhoff polytope.

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Cite

BibTeXRIS
BibTeX
@misc{deza2026facial,
  title = {{Facial distance and diameter}},
  author = {Antoine Deza and David Martínez-Rubio and Javier Peña and Elias Wirth},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32440},
  eprint = {2609.32440},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32440},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Facial distance and diameter
AU  - Antoine Deza
AU  - David Martínez-Rubio
AU  - Javier Peña
AU  - Elias Wirth
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32440
UR  - https://arxiv.org/abs/2609.32440
AB  - The facial distance of a polytope, equivalently its pyramidal width, is a geometric condition measure that appears in the convergence bounds for Frank--Wolfe algorithms. Peña and Wirth developed calculus rules for this quantity and conjectured that the ratio of the facial distance to the diameter multiplied by the square root of the dimension is bounded by a constant. We show that this quantity is uniquely maximized by a regular simplex in each dimension, which implies that the conjecture holds and that √2 is the smallest constant. We also compute the facial distance of the Birkhoff polytope.
ER  -