General (non-symmetric) logarithmic Minkowski problem

Determine necessary and sufficient conditions on a finite Borel measure μ on S^{n−1} to guarantee the existence of a convex body K in R^n (not assumed origin-symmetric) such that its cone-volume measure V(K,·) equals μ.

Background

The logarithmic Minkowski problem prescribes the cone-volume measure V(K,·) of a convex body K containing the origin. The symmetric case (even data) has been solved, but extending this to general measures without symmetry remains unresolved.

The authors highlight this gap in the context of their broader program connecting chord measures and cone-chord measures to Minkowski-type problems.