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.
