The Lp centro-sectional Minkowski problem

Abstract

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp dual Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.