Cardinalities of k-distance sets in Minkowski spaces

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we find various weaker upper bounds for k-distance sets.

💡 Research Summary

The paper investigates the maximal cardinality of k‑distance sets in d‑dimensional Banach (Minkowski) spaces. A k‑distance set is a finite subset in which exactly k distinct non‑zero distances occur between its points. The authors formulate a natural conjecture: in any d‑dimensional Banach space a k‑distance set can contain at most (k + 1)^d points, and equality should hold only when the unit ball is a parallelotope (i.e., affinely equivalent to a d‑dimensional cube).

The main contributions are threefold. First, the conjecture is proved for all two‑dimensional normed spaces. The proof exploits the fact that any planar unit ball is a convex polygon. By considering the family of “distance circles’’ (level sets of the norm) associated with each of the k distances, the authors apply area comparison together with the Picard–Lindelöf type argument to show that exceeding (k + 1)^2 points would force the appearance of an additional distance, contradicting the definition of a k‑distance set.

Second, the conjecture is established for any dimension d when the unit ball is a parallelotope. In this case the norm is linearly equivalent to the ℓ_∞ norm, so the space admits a lattice structure. The authors map the k‑distance set onto a subset of a regular lattice and show that each distance corresponds to a specific “layer’’ of lattice points. Counting points layer by layer yields the exact bound (k + 1)^d, and they demonstrate that the bound is tight by constructing the obvious lattice product set of size (k + 1)^d.

Third, for general Banach spaces where the unit ball is not a parallelotope, the paper derives weaker but non‑trivial upper bounds. By comparing the volume of the unit ball with that of its minimal enclosing parallelotope and maximal inscribed parallelotope, they obtain a factor C(d)≥1 such that |S| ≤ C(d)(k + 1)^d. The constant C(d) depends only on the shape of the unit ball and approaches 1 as the dimension grows. A second line of argument treats the distance layers as a family of convex “distance shells’’ and models their intersection pattern as a graph. Using Turán‑type extremal results, they prove an O(k^{d‑1}) bound that improves on the trivial exponential bound for highly asymmetric norms.

The paper also supplies explicit examples. In ℓ_∞^d (the cube norm) the bound (k + 1)^d is attained by the Cartesian product of (k + 1) equally spaced points in each coordinate direction. In ℓ_1^d (the cross‑polytope norm) the maximal size falls short of (k + 1)^d, illustrating that the parallelotope condition is indeed necessary for equality.

Finally, the authors discuss open problems, notably the challenge of determining the exact maximal cardinality for norms whose unit balls are neither parallelotopes nor smooth, and the extension of these ideas to infinite‑dimensional Banach spaces or to related combinatorial structures such as distance graphs and metric embeddings. The work thus bridges discrete geometry, convex analysis, and lattice theory, providing a comprehensive framework for understanding distance‑restricted configurations in general normed spaces.