A reverse isoperimetric inequality gives an upper bound for surface area in terms of volume after a suitable affine
transformation or under additional geometric restrictions. The ordinary isoperimetric
inequality instead gives a lower bound for surface
area at fixed volume. Write 
for a convex
body 
, where 
is its surface area and
is its volume. This convention uses surface area in the
numerator, rather than the convention for the isoperimetric
quotient.
Ball (1991) proved that every convex body has an image under an affine transformation whose value
of 
is no larger than that of a regular simplex. Among centrally symmetric convex
bodies, the corresponding extremal comparison is with a hypercube.
Su et al. (2026) proved that every convex body 
admits a linear transformation 
of determinant 1 such that,
for every 
, there is a convex
body 
satisfying
 |
and
 |
The same transformation 
works for all 
.
In particular, taking 
gives 
while retaining an inner half-sized copy
of 
.
