The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.

💡 Research Summary

The paper investigates a lattice‑constrained version of the classical isodiametric problem. In the unconstrained setting, among all sets of a given volume the Euclidean ball uniquely minimizes the diameter. Here the authors ask: if a centrally symmetric convex body K ⊂ ℝⁿ must avoid all non‑zero points of a fixed lattice L (i.e., int(K) ∩ (L \ {0}) = ∅), which shape attains the smallest possible diameter for a prescribed volume?

The main result shows that the extremal body is always the intersection of a suitably sized Euclidean ball B₍ᵣ₎ with the Dirichlet–Voronoi cell 𝔙(2L) of the doubled lattice 2L. More precisely, for any volume V there exists a unique radius r such that vol(B₍ᵣ₎ ∩ 𝔙(2L)) = V, and the set K* = B₍ᵣ₎ ∩ 𝔙(2L) has the minimal possible diameter among all centrally symmetric convex bodies of volume V that avoid interior lattice points of L. The proof proceeds in two stages. First, using Brunn–Minkowski theory together with Minkowski’s convex‑body theorem, the authors show that any admissible K can be translated and homothetically scaled into the Voronoi cell of 2L without increasing its diameter. Second, a symmetrisation (or “compression”) argument replaces K by the intersection of a ball with the same Voronoi cell, preserving volume while not increasing diameter. This establishes optimality of the ball‑Voronoi intersection.

Beyond existence, the authors conjecture that K* is the only extremal body for any lattice L and any dimension. They prove this uniqueness in three dimensions by a detailed classification of three‑dimensional Voronoi cells and a case‑by‑case analysis of possible contact configurations with the lattice. Moreover, the conjecture is verified for several highly symmetric lattices: the integer lattice ℤⁿ, the root lattices Aₙ and Dₙ, the exceptional lattice E₈ in dimension eight, and the Leech lattice in dimension twenty‑four. In each of these cases the high degree of symmetry forces any extremal body to inherit the same symmetry, which forces it to coincide with the ball‑Voronoi intersection.

The paper situates its findings within the broader context of lattice geometry, packing and covering problems, and convex optimization. By introducing lattice‑point avoidance as a natural constraint, the authors reveal that the classical ball is no longer optimal; instead the geometry of the underlying lattice dictates the shape of the optimal set. The result bridges two classical topics: the isodiametric inequality and the theory of Voronoi cells, and suggests new avenues for research. Potential extensions include: (i) removing the central‑symmetry assumption, (ii) considering multiple lattices or more general discrete sets, (iii) investigating analogous problems for other shape functionals such as surface area or mean width, and (iv) developing algorithmic methods to compute the optimal radius r for arbitrary lattices in high dimensions. The techniques introduced—particularly the combination of Brunn‑Minkowski compression with Voronoi‑cell analysis—are likely to be useful in these future directions.

In summary, the authors establish that for centrally symmetric convex bodies avoiding interior lattice points, the minimal‑diameter shape of prescribed volume is precisely the intersection of a Euclidean ball with the Voronoi cell of the doubled lattice. They confirm the uniqueness of this extremal shape in dimension three and for several prominent lattices, thereby opening a new line of inquiry at the intersection of convex geometry and lattice theory.