NP-hardness for lattice polytopes in dimensions at least three

Prove that, for every dimension d≥3, computing the lattice diameter of a lattice d-polytope is an NP-hard problem.

Background

The paper proves NP-hardness of computing lattice diameters for bounded semi-algebraic sets in every fixed dimension d≥3, even when the diameter direction is prescribed. The authors then conjecture that the same hardness persists when the input is restricted to lattice polytopes, a substantially narrower class of geometric objects.

References

Furthermore, we conjecture that this computation stays hard, even for lattice polytopes:

— On Lattice Diameter Segments and A Discrete Borsuk Partition Problem  (2508.20009 - Brose et al., 27 Aug 2025) in Remark following Theorem 2.7, Section 2.2