A sphere order is a partially ordered set which can be represented by closed balls in some finite-dimensional
Euclidean space, with 
iff the ball
representing 
is contained in the ball representing 
. For balls with centers 
and radii 
, this condition is
 |
where 
denotes the Euclidean norm.
Meyer (2026) proved that the Boolean lattice 
is not a sphere order, in any finite
dimension. Thus not every finite partially
ordered set admits such a representation.
Is Not a Sphere Order." 1 Oct 2026.