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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 }.

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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., 3 Jun 2024) in Section 9 (Bourgain’s slicing problem), bullet list of related conjectures