Minkowski problem for surface area measures of C-pseudo-cones (general case)

Determine necessary and sufficient conditions for a Borel measure on the spherical set ΩC◦ = S^{n−1} ∩ int(C◦) to be the surface area measure Sn−1(K,·) of a C-pseudo-cone K ∈ ps(C), without assuming finiteness or compact support of the measure.

Background

The paper studies pseudo-cones K in ℝ^n with recession cone C, focusing on measures defined on directions u in ΩC◦ = S^{n−1} ∩ int(C◦). For C-pseudo-cones, the surface area measure Sn−1(K,·) is a Borel measure defined on ΩC◦ that may be infinite and is only defined on this subset of the sphere.

Existing results show sufficiency under additional assumptions: when the Borel measure is finite and has compact support, there exists a C-pseudo-cone realizing it as a surface area measure (as established in prior work [21], with further Lp extensions and dual curvature analogues). However, a complete characterization—necessary and sufficient conditions in the absence of finiteness/compact support—has not been obtained, and this is noted explicitly as unknown in the introduction.