Minkowski problem for cone-volume measures of C-pseudo-cones without finiteness

Determine necessary and sufficient conditions for a (possibly infinite) Borel measure on ΩC◦ = S^{n−1} ∩ int(C◦) to be the cone-volume measure VK of some C-pseudo-cone K ∈ ps(C), i.e., characterize those measures that arise as cone-volume measures when the finiteness condition is dropped.

Background

The cone-volume measure VK of a C-pseudo-cone K is a Borel measure defined on ΩC◦, analogous to the convex body case but potentially infinite. Prior work [21] established that every nonzero finite Borel measure on ΩC◦ is a cone-volume measure of some C-pseudo-cone, and uniqueness was proved in [28]. This paper provides a necessary growth condition (Theorem 1) and later solves the weighted version completely (Theorem 2), where finiteness of the weighted measure is both necessary and sufficient.

Despite these advances, the unweighted Minkowski problem—characterizing all Borel measures (not necessarily finite) that are cone-volume measures of C-pseudo-cones—remains explicitly open when finiteness is not assumed.