Kusner conjecture

In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of $d$ -dimensional Euclidean space is $d+1$, achieved by the vertices of a regular simplex, and the equilateral dimension of a $d$ -dimensional vector space with the Chebyshev distance ($L^{\infty }$ norm) is $2^{d}$, achieved by the vertices of a hypercube. However, the equilateral dimension of a space with the Manhattan distance ($L^{1}$ norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly $2d$, achieved by the vertices of a cross polytope.