Abstract: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.
| Subjects: | Functional Analysis (math.FA) |
| Cite as: | arXiv:2605.25666 [math.FA] |
| (or arXiv:2605.25666v1 [math.FA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25666 arXiv-issued DOI via DataCite (pending registration) |
From: Youjiang Lin [view email]
[v1]
Mon, 25 May 2026 10:15:34 UTC (32 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。