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General $L_p$ dual curvature measures have recently been introduced byLutwak, Yang and Zhang. These new measures unify several other geometricmeasures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory.$L_p$ dual curvature measures arise from $q$th dual inrinsic volumes by meansof Alexandrov-type variational formulas. Lutwak, Yang and Zhang formulated the$L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dualcurvature 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 regularityof the solution.

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