Minimum-volume and minimum-vertex examples

Find a lattice polytope of minimum normalized volume, and a lattice polytope with the minimum number of vertices, that has a flag unimodular triangulation but no quadratic triangulation.

Background

The smallest example found in the census has 13 vertices and normalized volume 29. The authors leave unresolved whether these two measures are minimal, separately, among all lattice polytopes exhibiting a flag unimodular triangulation without a quadratic triangulation.

References

Find a lattice polytope of minimum normalized volume, and one with the minimum number of vertices, that has a flag unimodular triangulation but no quadratic triangulation.

— Most $(0,1)$-polytopes are not normal  (2609.02778 - Morales, 2 Sep 2026) in Unnamed Problem following the low-dimensionality problem, Section 5, Open problems