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Konoike’s question on magic positivity of polar duals

For every dd-dimensional reflexive lattice polytope P⊂RdP\subset \mathbb{R}^d, is the Ehrhart polynomial of its polar dual P∨={y∈Rd:⟨x,y⟩≤1 for all x∈P}P^\vee=\{y\in\mathbb{R}^d:\langle x,y\rangle\leq 1\text{ for all }x\in P\} magic positive? Equivalently, are all magic coefficients of LP∨(m)=∣mP∨∩Zd∣L_{P^\vee}(m)=|mP^\vee\cap\mathbb{Z}^d| nonnegative?

References

Primary source

arXiv

Additional references

combinatoricsehrhart-theorypolyhedral-geometry
Posted · edited

Progress summary

Refreshed
Claimed progress

A 2026 preprint claims major cases of the question are settled, but the general question remains open.

Konoike’s question asks whether a positivity property holds for polar duals of the relevant polytopes. The latest work reports substantial progress in two structured classes, without settling the broader conjecture.

Known results

  • Magic positivity was proved for the Ehrhart polynomials of Stasheff polytopes.
  • A partial proof was given for magic positivity of certain dual polytopes.

October 2026 preprint

Ohsugi and Tsuchiya claim magic positivity for pseudo-symmetric simplicial reflexive polytopes and strict positivity for pseudo-symmetric smooth Fano polytopes. The results imply real-rootedness and gamma-positivity consequences for associated h∗h^*-polynomials, but the preprint does not settle the broader question.

Current status (as of October 2026): magic positivity is claimed for pseudo-symmetric simplicial reflexive polytopes, with strict positivity claimed for pseudo-symmetric smooth Fano polytopes; the broader Konoike question remains open and these claims are unverified.

Sources

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