Equilateral dimension

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.^[1] Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages.^[1] The equilateral dimension of ${\displaystyle d}$-dimensional Euclidean space is ${\displaystyle d+1}$, achieved by the vertices of a regular simplex, and the equilateral dimension of a ${\displaystyle d}$-dimensional vector space with the Chebyshev distance (${\displaystyle L^{\infty }}$ norm) is ${\displaystyle 2^{d}}$, achieved by the vertices of a hypercube. However, the equilateral dimension of a space with the Manhattan distance (${\displaystyle L^{1}}$ norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly ${\displaystyle 2d}$, achieved by the vertices of a cross polytope.^[2]