The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
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发表时间：
2018-02
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通讯作者：
K. Boroczky;F. Fodor
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作者：
K. Boroczky;F. Fodor
一般的$L_p$对偶曲率度量最近由Lutwak，Yang和Zhang引入。这些新的度量统一了Brunn-Minkowski理论和对偶Brunn-Minkowski理论的其他几个几何度量。通过Alexandrov型变分公式，$L_p$对偶曲率度量由$q$次对偶本征体积得到。Lutwak，Yang和Zhang提出了$L_p$对偶Minkowski问题，它涉及到$L_p$对偶曲率测度的刻画。本文解决了$p>；1$和$q>；0$的$L_(P)$对偶Minkowski问题的存在部分，并讨论了解的正则性。
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.