Uniqueness and Continuity of the Solution to $L_p$ Dual Minkowski Problem

Abstract

Lutwak et al. (Adv Math 329:85–132, 2018) 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 Lutwak et al. (Adv Math 329:85–132, 2018), we consider the uniqueness and continuity of the solution to the $L_p$ dual Minkowski problem. To extend the important work (Theorem A) 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.

Keywords

$L_p$ dual Minkowski problem / $L_p$ dual curvature / Minkowski-type inequlity / Uniqueness / Continuity / 52A40

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