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Fixed points of the polar projection body operator and its square

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

Contributed by Sihan Chen ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Let TK=(ΠK)∘TK=(\Pi K)^\circ be the polar projection body of a convex body KK in Rn\R^n. In dimension n≥3n\ge3, we prove that TKTK is a positive dilate of KK exactly when KK is an origin-centred ellipsoid, settling Petty's polar projection conjecture. The same classification holds under the weaker assumption that T2KT^2K is a positive dilate of KK. The proof uses a distributional Laplacian identity to compare two integrals of central section volumes. Equality supplies contractive hyperplane projections with every kernel direction, and James's theorem gives ellipsoidal rigidity. The argument applies to arbitrary convex bodies and yields the translative constant volume conjecture of Horv\'ath and L\'angi.

Provenance statement

The entire derivation has been verified and re-derived by the author, and no issues were found. The author coincidentally solved the problem while adapting the ideas from the manuscript on the classification of fixed points of the center of mass; the main line of reasoning has been placed in the appendix. The final key derivation and ideas were completed by AI.

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References

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Version history

  1. v1 · 5 pagesSubmitted by Sihan ChenInitial depositCurrentNov 12, 2026