The $Lp$ chord Minkowski problem for negative $p$

TL;DR

This paper addresses the $L_p$ chord Minkowski problem for negative $p$, providing solutions for discrete measures and certain Borel measures, advancing the understanding of convex geometric measures.

Contribution

It offers the first solutions to the $L_p$ chord Minkowski problem for negative $p$ in discrete and specific Borel measure cases, extending prior work.

Findings

Solved for discrete measures in general position for negative $p$ and $q>0

Provided proof for Borel measures with density under certain conditions

Extended the $L_p$ Minkowski problem to include the chord Minkowski problem

Abstract

In this paper, we solve the $L_p$ chord Minkowski problem in the case of discrete measures whose supports are in general position for negative $p$ and $q>0.$ As for general Borel measure with a density, we also give a proof but need $p\in(-n,0)$ and $n+1>q\geqslant 1.$ The $L_p$ 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 $L_p$ chord measure of a convex body, and it includes the chord Minkowski problem and the $L_p$ Minkowski problem.