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Level C · Reviewed Hard Geometry P-unimodular-triangulation-dilation-constant

A uniform dilation constant for unimodular triangulations

Is there c_d such that c_d·P has a unimodular triangulation for every lattice d-polytope P? Knudsen–Mumford–Waterman give a dilation depending on P.

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Cite 
@misc{cairn-unimodular-triangulation-dilation-constant,
  title        = {A uniform dilation constant for unimodular triangulations},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/unimodular-triangulation-dilation-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
}

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The problem

The question

A lattice polytope has a unimodular triangulation if it can be triangulated into lattice simplices of normalised volume 1. Knudsen, Mumford and Waterman proved that for every lattice polytope P some dilation kP has a unimodular triangulation, with k depending on P. Is there a constant c_d, depending only on the dimension, such that c_d·P has a unimodular triangulation for every lattice polytope P in R^d?

What counts as progress

  • Proofs in low dimension or for classes of polytopes.
  • Lattice 4-polytopes for which small dilations have no unimodular triangulation — lower bounds on c_4, found by computer search (each case a finite check).

Source. Posed by Gaku Liu in an extended abstract of the Oberwolfach workshop Discrete Geometry (2024), recorded in Oberwolfach Reports 3/2024, p. 162 (EMS Press, DOI 10.4171/OWR/2024/3), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.