The Mahler volume product conjecture asserts that every convex body 
satisfies
 |
(1)
|
where 
is the Santaló point and the superscript
denotes the polar body. The equality case is conjectured
to be precisely the simplex. For convex
bodies symmetric about the origin, the symmetric version
asserts
 |
(2)
|
with equality precisely for images of Hanner polytopes under invertible linear transformations.
Winter (2026) proved that for a full-dimensional convex polytope 
symmetric about the origin,
 |
(3)
|
where 
is the number of maximal chains of nonempty faces. If the
symmetric Mahler lower bound holds, this inequality
gives 
,
the inequality in the Kalai flag conjecture,
but does not establish its equality classification.
The one-dimensional symmetric inequality is an equality for centered intervals, whose volume product is 4.
