Uniqueness in the log-Minkowski problem for convex bodies

Establish whether solutions to the log-Minkowski problem for convex bodies are unique; specifically, determine whether a convex body in Euclidean space is uniquely determined by its cone-volume measure, i.e., whether for any prescribed cone-volume measure on the unit sphere there exists at most one convex body whose cone-volume measure equals the given measure.

Background

In the case p=0, the L_p Minkowski problems become the log-Minkowski problems, which aim to characterize cone-volume measures of convex bodies. While existence results are known in various settings, the question of uniqueness has remained difficult in the convex bodies framework.

The authors note that, in contrast, for C-coconvex sets the uniqueness can be obtained via a log-Minkowski inequality, highlighting a qualitative difference between convex bodies and C-coconvex sets. The cited literature underscores the longstanding and challenging nature of the uniqueness question in convex geometry.