It is shown that within the $L_p$-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural $L_p$ extension, for all real $p$. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the $L_p$-integral curvature of a convex body. This problem is solved for positive $p$ and is answered for negative $p$ provided the given measure is even.
The first author was supported by the National Science Fund of China for Distinguished Young Scholars (No. 11625103) and the Fundamental Research Funds for the Central Universities of China. The other authors were supported, in part, by USA NSF Grants DMS-1312181 and DMS-1710450.
"The $L_p$-Aleksandrov problem for $L_p$-integral curvature." J. Differential Geom. 110 (1) 1 - 29, September 2018. https://doi.org/10.4310/jdg/1536285625