The Kusner conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer $d\geq1$, let $\ell_1^d$ mean $\mathbb R^d$ with norm $\|x\|_1=\sum_{j=1}^d|x_j|$. An equilateral set $S\subseteq\mathbb R^d$ has a number $a>0$ such that $\|x-y\|_1=a$ for every two distinct $x,y\in S$. Let $e(\ell_1^d)$ be the largest possible cardinality of such a set. The $2d$ points $\{\pm e_j:1\leq j\leq d\}$, where $e_j$ are the standard coordinate vectors, have all pairwise distances two, so $e(\ell_1^d)\geq2d$.

Statement recorded in the supplied source.

Does every equilateral set in $\ell_1^d$ have at most $2d$ points, for every $d\geq1$? Equivalently, is $e(\ell_1^d)=2d$? [S2]

Short English statement

Can there ever be more than twice the dimension many mutually equidistant points in a real space with the sum-of-coordinate-distances metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200051 (AMR-071-0051). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Konrad J. Swanepoel, Equilateral sets in finite-dimensional normed spaces (2004). Sections 1.1–1.2, Problem 1 (Kusner), pp. 2–3. https://arxiv.org/pdf/math/0406264v1.