MathDB

Mahler conjecture and Kalai’s flag conjecture

For every dd-dimensional centrally symmetric convex polytope P⊂RdP\subset\mathbb{R}^d, the number F(P)\mathcal{F}(P) of flags of PP satisfies F(P)≥2dd!\mathcal{F}(P)\ge 2^d d!. Hanner polytopes are conjectured to be extremal.

References

Primary source

arXiv

Additional references

convex-geometrygeometric-inequalitiespolytope-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

A new result links the two conjectures through a proved inequality, but neither conjecture itself has been solved.

Mahler’s conjecture and Kalai’s flag conjecture both predict Hanner polytopes as extremal objects. The general statements remain open.

Known results

  • Proper locally anti-blocking polytopes satisfy Kalai’s flag bound, with equality characterized by generalized Hanner polytopes.
  • Unconditional and locally anti-blocking cases satisfy Kalai’s 3d3^d face bound.
  • Mahler’s conjecture is known for hyperplane sections of ℓp\ell_p-balls and Hanner polytopes, and corresponding projections.
  • The flag bound remains conjectural for general centrally symmetric polytopes.

October 2026 inequality

On October 7, 2026, Martin Winter reported the inequality vol⁡(P)vol⁡(P−P)∘≤F(P)/(d!)2\operatorname{vol}(P)\operatorname{vol}(P-P)^\circ\le F(P)/(d!)^2, together with a centrally symmetric consequence. This is a substantive link between the conjectures, but it does not prove either one.

Current status (as of October 2026): The general Mahler and Kalai flag conjectures remain open; Winter’s new inequality is claimed progress, not a resolution.

Sources

Solutions 0

No solutions have been posted yet.