An equilateral set in a metric space is a set in which the distance between every pair of distinct points has the same positive
value. In a real normed space 
, this means 
for distinct 
in the set (Petty 1971).
In Euclidean 
, a regular simplex has
vertices and gives a largest equilateral
set. For the maximum norm 
, all
vertices of 
instead form an equilateral set of common distance
2. The Petty equilateral-set conjecture
asks whether every real 
-dimensional
normed space contains at least 
equidistant points. This is an existence question, rather
than a statement that every equilateral set extends to that size.
Over a field 
of field characteristic
other than 2, the algebraic version uses 
and requires 
to have the same nonzero value for every distinct pair.
This quadratic form need not define a metric.
Swanepoel (2026) proved that in field characteristic
zero an equilateral set of size 
exists iff 
is even and 
is a square in 
, or 
, or 
and 
is a sum of two squares in 
. The case 
is the classical regular-simplex
theorem of Schoenberg (1937).
If 
has odd field characteristic 
, a set of size 
exists iff 
divides 
and either 
is odd or 
is a square in 
. No such equilateral set has
more than 
elements. In a set of size 
, each point
is the geometric centroid of the other 
points (Swanepoel
2026).
