Fixed Point Rigidity of the Operator ΓpΠp∗ and the LYZ Conjecture
Abstract
We characterize the fixed points of the operator ΓpΠp∗ for n≥ 3 and 1<p<n. More precisely, we prove that a convex body K∈𝒦on satisfies ΓpΠp∗ 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 Lp setting, we introduce the Lp-Projection Rolodex, which provides a dimensional reduction of the volume of the polar Lp-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 voln(Πp∗ Kt) to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.
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BibTeX
@misc{lin2026fixed,
title = {{Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture}},
author = {Youjiang Lin and Sudan Xing},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.25666},
eprint = {2605.25666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25666},
}
RIS
TY - UNPB
TI - Fixed Point Rigidity of the Operator Γ_pΠ_p^∗ and the LYZ Conjecture
AU - Youjiang Lin
AU - Sudan Xing
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2605.25666
UR - https://arxiv.org/abs/2605.25666
AB - We characterize the fixed points of the operator Γ_pΠ_p^∗ for n≥ 3 and 1<p<n. More precisely, we prove that a convex body K∈𝒦_o^n satisfies Γ_pΠ_p^∗ 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 vol_n(Π_p^∗ K_t) to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.
ER -