Uniqueness in the logarithmic Minkowski problem for cone-volume measure

Establish whether a convex body K containing the origin is uniquely determined by its cone-volume measure V(K,·); that is, prove or refute that V(K_1,·) = V(K_2,·) implies K_1 = K_2 under the standard assumptions of the logarithmic Minkowski problem.

Background

In addition to existence, uniqueness is a central aspect of the logarithmic Minkowski problem. While existence is known in the symmetric case, the question of uniqueness remains unresolved in general.

The authors explicitly note the status of uniqueness as open and connect developments around it to broader conjectures (notably the log-Brunn–Minkowski inequality).