Fixed Point Rigidity of the Operator ${\Gamma }_p{\Pi }_p^{\ast }$ and the LYZ Conjecture

Abstract

We characterize the fixed points of the operator ${\Gamma }_p{\Pi }_p^{\ast }$ for $n\ge 3$ and $1<p<n$. More precisely, we prove that a convex body $K\in {\mathcal{K}}_o^n$ satisfies \[Γ_pΠ_p^\ast K=cK\] for some $c>0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces ${\mathrm{vol}}_n({\Pi }_p^{\ast }{K}_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.