The logarithmic Brunn-Minkowski inequality compares the volume of the logarithmic interpolation of two convex bodies
with the geometric mean of their volumes. For 
containing the origin
in their interiors and 
, define
 |
where 
and 
are the support functions.
The inequality is
 |
Liu et al. (2026) reported a proof for all 
when both centroids
are at the origin, without assuming central symmetry.
Equality holds iff there are complementary subspaces 
,
..., 
, centered convex bodies 
,
and positive scalars 
such that 
and 
. Here direct sums
of sets are Minkowski sums in complementary subspaces,
which need not be orthogonal.
Liu et al. (2026) state that all proofs were generated by ChatGPT and were subjected to adversarial checks. Independent verification of the reported proof is not asserted here.
