Universal cn(log n)^{1+log_2 e} upper bound for lattice covering density of all convex bodies

Establish that there exists an absolute constant c > 0 such that for every integer n ≥ 1 and every n-dimensional convex body K in Euclidean space R^n, the lattice covering density θ_L(K) (the density of the thinnest lattice covering of R^n by translates of K) satisfies θ_L(K) ≤ c n (log_e n)^{1 + log_2 e}.

Background

The paper studies lattice covering densities θ_L(K) for various classes of convex bodies. Classical results include Rogers’s general upper bounds and the improved bound θ_L(K) ≤ c n2 by Ordentlich–Regev–Weiss (2021). For special bodies, Rogers obtained θ_L(Bn) ≤ c n (log n){(1/2) log_2(2πe)}, and Gritzmann extended near-linear bounds to certain symmetric convex bodies, proving θ_L(K) ≤ c n (log n){1 + log_2 e} when K has sufficient hyperplane symmetries.

This work proves that the same near-linear polylogarithmic bound θ_L(K) ≤ c n (log n){1 + log_2 e} holds for two broader classes without symmetry assumptions: all n-dimensional quarter-convex bodies and all n-dimensional polytopes with n+2 vertices. Motivated by these results and earlier bounds, the authors conjecture that this bound should hold universally for all convex bodies in Rn. Confirming this would significantly strengthen the general upper bound from c n2 to near-linear in n up to a logarithmic factor.

References

By (1)–(4) and Theorem 1–2, we make a conjecture that for all K∈𝒦n, θ_L(K)≤cn(log_e n){1+log_2 e} for some constant c.

— On lattice coverings by locally anti-blocking bodies and polytopes with few vertices (2505.07369 - Schymura et al., 12 May 2025) in Remark, Section 1 (Introduction)