Horizontal perimeter is the total variation of the horizontal distributional gradient of the characteristic function of a measurable set in a space with a specified horizontal orthonormal basis of vector fields.
For the first Heisenberg group in coordinates 
with 
and 
, it is
 |
where the bars denote total variation, 
is the characteristic
function of 
,
and 
and 
are distributional derivatives with respect to Lebesgue
measure (Chen et al. 2026). Equivalently, it is the supremum of 
over smooth,
compactly supported pairs 
satisfying 
.
The Pansu conjecture concerns minimizers of horizontal perimeter at fixed volume.
