An isoperimetric profile gives the least possible perimeter of a set with a prescribed volume.
For an open set 
, its relative isoperimetric
profile is
 |
where 
is relative perimeter and 
is Lebesgue measure.
For the unit cube 
, Deng et al. (2026)
reported
 |
where 
and 
. Up to isometries
of the cube, complementation, and changes on sets
of measure zero, the minimizers are corner eighth-balls, edge quarter-cylinders,
and coordinate slabs. The transitions occur at 
and 
, with both adjacent types
minimizing at each transition (Deng et al. 2026).
