Intermediate Christoffel–Minkowski problem (general case)

Determine necessary and sufficient conditions for a Borel measure on the unit sphere S^{n−1} to be the i-th order area measure S_i(K,·) of some convex body K ⊂ R^n for the intermediate degrees 1 < i < n−1, without any symmetry or smoothness assumptions; that is, solve the intermediate Christoffel–Minkowski problem in full generality.

Background

Area measures S_i(K,·) encode local geometric information of a convex body K and generalize surface area measures. The classical Christoffel–Minkowski program seeks to characterize which Borel measures on S^{n−1} arise as S_i(K,·).

Complete solutions are known for i=1 (Christoffel’s problem, resolved by Berg and Firey) and for i=n−1 (Minkowski–Alexandrov’s theorem). For 1<i<n−1, Guan–Ma provided a rich sufficient condition, but a full characterization is unknown.

The present paper solves a broad class of anisotropic and mixed Christoffel–Minkowski problems under axial symmetry (bodies of revolution), thereby refining earlier zonal results; however, the unrestricted intermediate case remains open.