Let K⊂Rn be a bounded convex body. We prove a lattice-averaging formula that gives upper bounds for the number of unit balls required to cover K. If the Voronoi cell P of a lattice is contained in the Euclidean unit ball, then some translate and rotation of the lattice produces a covering whose size is at most a linear combination of the intrinsic volumes of K; the coefficients are determined by the intrinsic volumes of P. The proof averages the number of Voronoi cells meeting K over one fundamental cell and over SO(n). For the regular hexagonal lattice in R2, the formula reproduces the classical planar Blaschke bound. For the cubic lattice, it gives a closed-form estimate in every dimension n≥2. In R3, explicit computations for the cubic, face-centered cubic, and body-centered cubic Voronoi cells show that the body-centered cubic lattice has the smallest coefficientwise bound among these three lattices. In R4, the intrinsic volumes of the A4∗ permutohedron are computed from its graphical-zonotope representation, leading to a sharper bound than for the cubic lattice.
Elad Atia (2026) studied this question.