The \(L_p\) Chord Minkowski Problem for Negative p

In this paper, we solve the \(L_p\) $L_p$ chord Minkowski problem in the case of discrete measures whose supports are in general position for \(p<0\) $p<0$ and \(q>0.\) $q>0.$ As for general Borel measure, we also give a proof but requiring \(-n< p<0\) $-n<p<0$ and \(n+1>q\geqslant 1.\) $n+1>q⩾1.$ The \(L_p\) $L_p$ chord Minkowski problem was recently posed by Lutwak, Xi, Yang, and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure, such that it is the \(L_p\) $L_p$ chord measure of a convex body. Moreover, The \(L_p\) $L_p$ chord Minkowski problem includes the chord Minkowski problem and the \(L_p\) $L_p$ Minkowski problem.