The $L_p$ dual Minkowski problem for $p>1$ and $q>0$

Abstract

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. $L_p$ dual curvature measures arise from $q$th dual inrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_{p}$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.