Uniqueness and continuity of the solution to $L_p$ dual Minkowski problem

TL;DR

This paper investigates the uniqueness and continuity of solutions to the $L_p$ dual Minkowski problem, extending previous results to general convex bodies through new inequalities and analysis.

Contribution

It establishes the uniqueness and continuity of solutions for the $L_p$ dual Minkowski problem for general convex bodies, extending prior polytope results.

Findings

Uniqueness of solutions when q < p for general convex bodies.

Continuity of solutions with respect to convex bodies.

Development of new Minkowski-type inequalities related to the problem.

Abstract

Lutwak, Yang and Zhang \cite{LYZ2018} introduced the $L_p$ dual curvature measure that unifies several other geometric measures in dual Brunn-Minkowski theory and Brunn- Minkowski theory. Motivated by works in \cite{LYZ2018}, we consider the uniqueness and continuity of the solution to the $L_p$ dual Minkowski problem. To extend the important work (Theorem \ref{uniquepolytope}) of LYZ to the case for general convex bodies, we establish some new Minkowski-type inequalities which are closely related to the optimization problem associated with the $L_p$ dual Minkowski problem. When $q< p$, the uniqueness of the solution to the $L_p$ dual Minkowski problem for general convex bodies is obtained. Moreover, we obtain the continuity of the solution to the $L_p$ dual Minkowski problem for convex bodies.