Fixed Point Rigidity of the Operator $\Gamma_p\Pi_p^\ast$ and the LYZ Conjecture

Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \cite{MilmanShabelmanYehudayoff2025}, we establish, for $p\geq 1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, we prove that if a convex body $K \in \mathcal{K}_o^n$ satisfies $$\Gamma_p \Pi_p^* K = cK$$ for some constant $c>0$, then $K$ must be an ellipsoid. Conversely, ellipsoids are the only such fixed points of convex bodies up to dilation. This confirms a conjecture of Lutwak, Yang, and Zhang \cite{LutwakYangZhang2000} for $p\geq 1$. Our approach combines variational techniques with a refined analysis of linear reflection shadow systems. We introduce a geometric framework, called the $L_p$-projection Rolodex, that represents the volume of the polar $L_p$-projection body in terms of weighted lower-dimensional sections. This representation yields a monotonicity property of the volume $\operatorname{Vol}_n(\Pi_p^*K_t)$ along linear reflection shadow systems $K_t$ and leads to a rigidity statement showing that the vanishing of the first variation forces constancy along the deformation. These results, together with known characterizations of equality in Steiner symmetrization, give the desired classification of fixed points.