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Classification of fixed points of the centroid body operator

Primarily AI-generated textHuman understanding: some partsmath.MG — Metric Geometry

Contributed by Sihan Chen ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We classify the fixed points, up to dilation, of the classical volume-normalized centroid body operator in every dimension. We prove that ΓK = cK holds precisely for ellipsoids centered at the origin. The proof combines a signed spherical moment-gap argument in low dimensions with boundary integral operators, compact-group structure, and quantitative regression estimates in higher dimensions. Affine eigenfunctions force a large rotation subspace; curvature and log-concave regression bounds then exclude every proper such subspace. The remaining finite comparisons are certified by exact rational inequalities. The manuscript supplies proofs of the auxiliary analytic estimates and explicitly identifies the external results used.

Provenance statement

The overall idea and general framework of the proof have been re-derived and checked by the author, with no problems found, and the numerical calculations have also been verified. Further tracing of the ideas and simplification of the proof have not been completed. In this process, the author used an interactive exploratory approach: the AI provided options and possible subsequent routes, and the author thought about, selected, and tried them (by AI as well), while ruling out some less likely methods. The entire proof took about three days. During this period, the author temporarily gave up for a while because of a lack of ideas, but the AI provided a key idea concerning an explicit upper bound and reduced it to 500; then, by improving the estimate, the AI lowered it further, and finally completed the proof for all dimensions.

Tools used

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References

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Version history

  1. v1 · 30 pagesSubmitted by Sihan ChenInitial depositCurrentNov 11, 2026