
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the equilateral dimension of a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
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
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is
, achieved by the vertices of a regular
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, and the equilateral dimension of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
with the
Chebyshev distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a real coordinate space where the distance between two points is the greatest of their differences along any coordinate dimensio ...
(
norm) is
, achieved by the vertices of a
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
. However, the equilateral dimension of a space with the
Manhattan distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...
(
norm) is not known. Kusner's conjecture, named after
Robert B. Kusner, states that it is exactly
, achieved by the vertices of a
cross polytope.
Lebesgue spaces
The equilateral dimension has been particularly studied for
Lebesgue spaces, finite-dimensional
normed vector space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
s with the
norm
The equilateral dimension of
spaces of dimension
behaves differently depending on the value of
:
*For
, the
norm gives rise to
Manhattan distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian ...
. In this case, it is possible to find
equidistant points, the vertices of an axis-aligned
cross polytope. The equilateral dimension is known to be exactly
for
, and to be upper bounded by
for all
.
[.] Robert B. Kusner suggested in 1983 that the equilateral dimension for this case should be exactly
;
this suggestion (together with a related suggestion for the equilateral dimension when
) has come to be known as Kusner's conjecture.
*For