Extreme inequalities of general $L_p$ $\mu$-projection body and general $L_p$ $\mu$-centroid body

TL;DR

This paper introduces general $L_p$ projection and centroid bodies for measures with positive homogeneity density, establishing extreme inequalities, measure comparison results, and monotone inequalities to advance convex geometric analysis.

Contribution

It defines new $L_p$ bodies for general measures and proves novel extreme inequalities and measure comparison results, extending classical convex geometric concepts.

Findings

Established extreme inequalities for general $L_p$ projection and centroid bodies.

Proved measure comparison theorems for these new bodies.

Derived monotone inequalities related to the measure properties.

Abstract

In this paper, we introduce the concept of general $L_p$ projection body and general $L_p$ centroid body of general measures with positive homogeneity density function, and prove the corresponding extreme inequalities. Meanwhile, we also study their measure comparison problem and monotone inequalities.