A positive solution to the $L^p$ projection centroid conjecture

TL;DR

This paper provides a positive solution to the longstanding $L^p$ projection centroid conjecture, advancing the understanding of $L^p$ geometric inequalities.

Contribution

It offers the first confirmed proof of the $L^p$ projection centroid conjecture, resolving a key open problem in convex geometry.

Findings

Confirmed the $L^p$ projection centroid conjecture.

Extended the $L^p$ analogs of classical geometric inequalities.

Abstract

In a classical paper [20] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [20], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.