Nonuniqueness of solutions to the $L_p$ chord Minkowski problem

TL;DR

This paper investigates the nonuniqueness of solutions to the $L_p$ chord Minkowski problem for negative $p$, revealing that solutions are not always unique under certain conditions.

Contribution

It demonstrates the nonuniqueness of solutions to the $L_p$ chord Minkowski problem for negative $p$, expanding understanding of this geometric measure problem.

Findings

Solutions are not unique for negative $p$ in the $L_p$ chord Minkowski problem.

The problem includes and generalizes the chord Minkowski and $L_p$ Minkowski problems.

Provides conditions under which nonuniqueness occurs.

Abstract

This paper explores the nonuniqueness of solutions to the $L_p$ chord Minkowski problem for negative $p.$ 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.