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arxiv: 2606.04774 · v1 · pith:RDREJETKnew · submitted 2026-06-03 · 🧮 math.AP · math.DG

Weighted centro-affine Poincar\'e inequalities

Yingxiang Hu , Mohammad N. Ivaki This is my paper

Pith reviewed 2026-06-28 05:22 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords centro-affine Poincaré inequalityweighted inequalitiesL0-Brunn-Minkowski inequalityunconditional convex bodiesdual quermassintegralMinkowski problem
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The pith

A flat logarithmic centro-affine geometry on the positive orthant yields sharp weighted Poincaré inequalities for unconditional convex bodies.

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

The authors introduce a flat logarithmic centro-affine geometry on the positive orthant that respects the multiplicative structure of the L0-sum. This geometry produces weighted Bochner formulas leading to Poincaré inequalities on spherical caps and their intersections for weights depending on the position vector. In the unconditional setting, it delivers a Poincaré inequality with weight |X|^2 and furnishes a new proof of the centro-affine Poincaré inequality with constant n. These results imply Brunn-Minkowski inequalities for dual quermassintegrals and uniqueness statements for certain Minkowski problems.

Core claim

The paper establishes that a flat logarithmic centro-affine geometry on (0,∞)^n, adapted to the L0-sum, yields via a Bochner formula a sharp Poincaré inequality, together with a new proof of the centro-affine Poincaré inequality with constant n for unconditional bodies and unconditional functions.

What carries the argument

flat logarithmic centro-affine geometry on the positive orthant adapted to the multiplicative L0-sum

If this is right

  • Weighted Poincaré inequalities hold on caps and intersections of caps for position-dependent weights.
  • A centro-affine Poincaré inequality with weight |X|^2 is obtained in the unconditional case.
  • Brunn-Minkowski inequalities are proved for the (n+2)nd dual quermassintegral and for the q-th dual quermassintegral when q in (0,n).
  • L_p-Brunn-Minkowski inequalities hold for q = n + alpha with alpha up to 2p(1-p)/(2-p).
  • These inequalities imply uniqueness results for the L_{p,q}-Minkowski problem in the unconditional class.

Where Pith is reading between the lines

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

  • The geometry might extend to yield Poincaré inequalities in other multiplicative settings beyond unconditional bodies.
  • Similar flat geometries could be constructed for other affine invariants in convex geometry.
  • Applications to functional inequalities for log-concave measures on the orthant may follow from the Bochner formula.

Load-bearing premise

The approach assumes the hypersurfaces are smooth and strictly convex and restricts attention to unconditional bodies and functions on the positive orthant.

What would settle it

Constructing an unconditional function on the positive orthant for which the centro-affine Poincaré inequality with constant n fails would disprove the result.

read the original abstract

We obtain weighted centro-affine Bochner formulas on spherical caps associated with smooth strictly convex hypersurfaces. As a consequence, we prove weighted Poincar\'e inequalities on caps and on intersections of caps for a class of weights depending on the position vector $X$ of the hypersurface. In the unconditional case, we obtain a centro-affine Poincar\'e inequality with weight $|X|^2$, which is used to prove a Brunn--Minkowski inequality for the $(n+2)$-nd dual quermassintegral. We also establish an $L_0$-Brunn--Minkowski inequality for the $q$-th dual quermassintegral for $q\in(0,n)$, with equality only for dilates, and an $L_p$-Brunn--Minkowski inequality for $q=n+\alpha$ whenever \[ 0<\alpha\le \frac{2p(1-p)}{2-p}, \] which in particular covers the range $q\in(n,n+6-4\sqrt{2}]$ for suitable $p\in(0,1)$. These Brunn--Minkowski inequalities imply weighted centro-affine Poincar\'e inequalities and uniqueness results for the $L_{p,q}$-Minkowski problem in the unconditional class. Our main contribution is the introduction of a flat logarithmic centro-affine geometry on the positive orthant $(0,\infty)^n$, adapted to the multiplicative structure of the $L_0$-sum. In this geometry, a Bochner formula yields a sharp Poincar\'e inequality, as well as a new proof of the centro-affine Poincar\'e inequality with constant $n$ due to Kolesnikov--Milman, for unconditional bodies and unconditional functions.

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 introduces a flat logarithmic centro-affine geometry on the positive orthant (0,∞)^n adapted to the multiplicative L0-sum structure. From this geometry it derives weighted centro-affine Bochner formulas on spherical caps of smooth strictly convex hypersurfaces. These formulas yield weighted Poincaré inequalities on caps and cap intersections for position-dependent weights |X|^2. In the unconditional setting the construction recovers the centro-affine Poincaré inequality with sharp constant n (Kolesnikov–Milman) and produces an L0-Brunn–Minkowski inequality for the q-th dual quermassintegral (q∈(0,n)) together with an Lp-Brunn–Minkowski inequality for q=n+α when 0<α≤2p(1-p)/(2-p). The inequalities imply uniqueness results for the unconditional L_{p,q}-Minkowski problem.

Significance. If the flatness of the introduced connection and the ensuing Bochner identity hold as stated, the work supplies a geometrically natural framework that simultaneously produces sharp weighted Poincaré inequalities and new Brunn–Minkowski statements for dual quermassintegrals. The explicit recovery of the constant n and the parameter ranges for the Lp case constitute concrete, falsifiable advances in centro-affine convex geometry.

minor comments (2)
  1. The abstract states the range 0<α≤2p(1-p)/(2-p) without indicating its origin; a one-sentence pointer to the relevant curvature computation or inequality in the body would improve readability.
  2. The precise definition of the flat logarithmic centro-affine connection (vanishing curvature, Christoffel symbols, etc.) should appear in a dedicated subsection before the Bochner formula is invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the detailed summary, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new flat logarithmic centro-affine geometry on (0,∞)^n adapted to the L0-sum, then derives a Bochner formula from the vanishing curvature of this connection; the weighted Poincaré inequalities and the reproof of the Kolesnikov–Milman result with constant n are direct consequences of that formula. No step equates the target inequality to a fitted parameter, renames a known result, or reduces the central claim to a self-citation whose content is itself unverified. The construction is explicitly restricted to the stated setting (smooth strictly convex hypersurfaces, position-dependent weights, unconditional bodies) and is presented as an independent geometric input rather than being reverse-engineered from the desired inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on standard convexity assumptions, the unconditional restriction, and the newly introduced geometry; no free parameters or fitted constants are mentioned.

axioms (2)
  • domain assumption Hypersurfaces are smooth and strictly convex
    Required to define spherical caps and the centro-affine setting.
  • domain assumption Bodies and functions are unconditional
    Used for the unconditional case and the geometry on the positive orthant.
invented entities (1)
  • flat logarithmic centro-affine geometry no independent evidence
    purpose: Adapted to the multiplicative structure of the L0-sum to produce a Bochner formula and sharp Poincaré inequality
    New framework introduced by the authors; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5851 in / 1452 out tokens · 29034 ms · 2026-06-28T05:22:50.185877+00:00 · methodology

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Reference graph

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