A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.
Research of the first author supported, in part, by NSFC No.11371360; research of the other authors
supported, in part, by NSF Grant DMS-1312181.
"Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems." Acta Math. 216 (2) 325 - 388, 2016. https://doi.org/10.1007/s11511-016-0140-6