Minkowski Lattice Conjecture

Establish that for any lattice L ⊂ ℝ^n with determinant 1, every translate of L intersects the set { x ∈ ℝ^n : ∏_{i=1}^n |x_i| ≤ 1/2^n }.

Background

The Minkowski lattice conjecture is a classical problem in the geometry of numbers. The notes connect it to extremizers of the isotropic constant: if the cube maximizes the isotropic constant among centrally symmetric convex bodies, then the conjecture follows.

While proven in dimension 2, the general case remains open and has deep ties to covering/packing and transference principles.

References

The Minkowski lattice conjecture suggests that if L ⊂ ℝn is a lattice of determinant one, then each of its translates intersects the set { x ∈ ℝn ; ∏_{i=1}n |x_i| ≤ 1/2n }.

— Isoperimetric inequalities in high-dimensional convex sets  (2406.01324 - Klartag et al., 2024) in Section 9 (Bourgain’s slicing problem), bullet list of related conjectures

Minkowski has further conjectured that for totally real number fields of degree $n$ the inequality $M(K) \le 2{-n}\sqrt{d}$ holds; this has however hitherto been proved only in special cases ($n = 2, 3, 4, 5$; see on this Remak , Dyson and Skubenko ).

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Remarks on Section 2