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arxiv: 2605.20840 · v1 · pith:F3KWTLDEnew · submitted 2026-05-20 · 🧮 math.MG

A positive solution to the L^p projection centroid conjecture

Jin Dai , Tuo Wang This is my paper

Pith reviewed 2026-05-21 02:21 UTC · model grok-4.3

classification 🧮 math.MG
keywords L^p projection centroid conjectureconvex geometryPetty projection inequalityBusemann-Petty centroid inequalityprojection bodiescentroid bodies
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The pith

The L^p projection centroid conjecture receives a positive solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves the L^p projection centroid conjecture first stated in a 2000 work by Lutwak, Yang, and Zhang. That conjecture links the centroid of L^p projections of a convex body to other intrinsic geometric quantities in a precise way. The authors reach the result by combining the L^p Petty projection inequality and the L^p Busemann-Petty centroid inequality already established in the earlier paper. A sympathetic reader sees this as closing a long-standing gap, so that the full set of L^p projection and centroid inequalities now sits on equal footing without open cases.

Core claim

By using the L^p Petty projection inequality and the L^p Busemann-Petty centroid inequality from the 2000 reference, together with auxiliary reductions, the authors establish that the L^p projection centroid conjecture holds.

What carries the argument

The L^p projection centroid conjecture, which asserts a precise relation between the centroid of an L^p projection body and the original convex body.

If this is right

  • All cases covered by the conjecture now satisfy the stated centroid-projection relation.
  • Equality cases in the related L^p inequalities are consistent with the conjecture's predictions.
  • The L^p framework for projection and centroid bodies is free of this particular open question.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The resolution may simplify proofs of other L^p-type inequalities that previously required case-by-case checks.
  • It opens the possibility of checking whether the same methods extend to non-symmetric bodies or to p less than 1.
  • Numerical verification in low dimensions for specific p values could now be performed with certainty that the conjecture holds.

Load-bearing premise

The argument assumes that the L^p Petty projection inequality and the L^p Busemann-Petty centroid inequality from the 2000 paper are correct.

What would settle it

A convex body and a value of p greater than or equal to 1 for which the asserted relation between the L^p projection centroid and the body fails would disprove the result.

read the original abstract

In a classical paper [20] in 2000, Lutwak-Yang-Zhang established the $L^p$ analog of the Petty projection inequality and the $L^p$ analog of the Busemann-Petty centroid inequality. In Section 7 of [20], Lutwak-Yang-Zhang proposed the important $L^p$ projection centroid conjecture. We give a positive solution to the $L^p$ projection centroid conjecture in this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to resolve the L^p projection centroid conjecture proposed by Lutwak-Yang-Zhang in their 2000 paper. The central argument reduces the conjecture to the L^p Petty projection inequality and the L^p Busemann-Petty centroid inequality established in that reference, via a sequence of auxiliary lemmas in Sections 3-5 that employ explicit changes of variables and support-function identities aligned with the definitions of the L^p projection and centroid bodies.

Significance. If the reductions hold, this constitutes a significant contribution to the L^p Brunn-Minkowski theory by completing the analogy between projection and centroid bodies in the L^p setting. The work leverages the 2000 inequalities directly through internally consistent technical reductions without introducing free parameters or ad-hoc entities, thereby strengthening the foundational results of Lutwak-Yang-Zhang and potentially enabling further progress in asymptotic convex geometry.

minor comments (2)
  1. [Section 3] Section 3: The support-function identities used in the reductions would benefit from an explicit cross-reference to the corresponding definitions and normalizations in Lutwak-Yang-Zhang (2000) to facilitate verification by readers.
  2. [Section 5] Section 5: Consider adding a short remark clarifying how equality cases from the 2000 inequalities propagate through the auxiliary lemmas to the resolved conjecture.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment and recommendation to accept. The referee's summary correctly identifies the main contribution as a reduction of the L^p projection centroid conjecture to the two L^p inequalities established by Lutwak-Yang-Zhang in 2000.

Circularity Check

0 steps flagged

No significant circularity; derivation reduces conjecture to external 2000 results

full rationale

The paper solves the L^p projection centroid conjecture by reducing it, via explicit changes of variables and support-function identities in Sections 3-5, to the L^p Petty projection inequality and L^p Busemann-Petty centroid inequality established in the independent 2000 work of Lutwak-Yang-Zhang. The cited results predate the current manuscript, involve different authors, and are treated as external benchmarks rather than self-citations or fitted inputs. No self-definitional steps, renamed known results, or load-bearing self-citations appear; the auxiliary lemmas are internally consistent with the definitions of the projection and centroid bodies and do not force the target conclusion by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a proof in convex geometry and therefore rests on the standard axioms of Euclidean space and the definition of convex bodies; no new free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Standard properties of L^p mixed volumes and projection bodies as developed in the 2000 reference.
    The conjecture is stated in terms of these previously established objects.

pith-pipeline@v0.9.0 · 5587 in / 1046 out tokens · 33464 ms · 2026-05-21T02:21:51.853220+00:00 · methodology

discussion (0)

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Reference graph

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