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).
Deng et al. (2026) report mathematical assistance from ChatGPT with proof strategies and calculations.
