On integral boxes of minimal surface

Generalising the two-dimensional case, we provide estimates for the mean-values of the lengths of the edges of an integral box with given volume and minimal surface.

💡 Research Summary

The paper studies the problem of minimizing the surface area of a k‑dimensional “integral box’’ – that is, a rectangular box in ℤ⁺ⁿ whose edges d₁,…,d_k are positive integers and whose volume ∏{i=1}^k d_i equals a given integer n. The surface area of such a box is σ(d₁,…,d_k)=2∑{h=1}^k∑{m=1}^k d_h d_m. For each n the authors denote by ρ₁(n)≤…≤ρ_k(n) the ordered edge lengths that achieve the minimal surface; these satisfy the optimisation problem (P{n,k}). In dimension two the solution is classical: ρ₁·ρ₂=n with ρ₁≤√n≤ρ₂, and the pair is unique. For higher dimensions uniqueness is not proved in general, but numerical experiments suggest it holds for k∈{3,4,5} and n≤10⁸.

The main contributions are two asymptotic formulas for the average values of the ρ_h over n≤x.

Theorem 1.1 (smallest edge).
Define Q(v)=v log v−v+1 and δ_k=Q(k−1)/log k. Then for the smallest edge ρ₁, \