Uniqueness and continuity of the solution to L_p dual Minkowski problem

2021/03/24 by Hejun Wang, Wang, Hejun, Jiazu Zhou +1
Mathematics · Medicine · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2103.13075

openalex publication_date 2021/03/24 · openalex created_date 2024/02/09 · openalex updated_date 2026/07/28

Abstract

Lutwak, Yang and Zhang \citeLYZ2018 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 \citeLYZ2018, we consider the uniqueness and continuity of the solution to the L_p dual Minkowski problem. To extend the important work (Theorem \refuniquepolytope) 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.

Related

• The L_p dual Minkowski problem for p>1 and q>0
• Dual curvature measures on convex functions and associated Minkowski problems
• The L_p Dual Minkowski Problem for Group-Invariant Convex Bodies
• The L_p dual Minkowski problem for unbounded closed convex sets
• Existence of solutions to the generalized dual Minkowski problem