Describe polytopes attaining the sharp universal center

Determine a structural description of the lattice polytopes satisfying \(\tau(P)=\lfloor(d-1)/2\rfloor\), beyond the generalized Reeve simplices and their lattice pyramids.

Background

The paper proves that the least universal integral Taylor-nonnegativity and Taylor-positivity centers in dimension dd equal ⌊(d−1)/2⌋\lfloor(d-1)/2\rfloor. Generalized Reeve simplices and their pyramids demonstrate sharpness by attaining this value.

However, the paper does not determine which other lattice polytopes attain the bound or what intrinsic geometric or combinatorial features characterize all such extremal polytopes. This is explicitly listed among the remaining classification problems.

References

Several classification problems remain open. It would be useful to characterize natural classes for which \rho(P)=0, especially within the families of matroid base polytopes and smooth polytopes. One may also ask for a structural description of the polytopes attaining \tau(P)=\lfloor(d-1)/2\rfloor, beyond the generalized Reeve constructions and their pyramids.

— Taylor Positivity of Ehrhart Polynomials  (2609.03327 - Liu et al., 3 Sep 2026) in Section 10, Concluding remarks