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arxiv: 2607.00567 · v1 · pith:IJJLBLJ4new · submitted 2026-07-01 · 🧮 math.MG

L_p Brunn-Minkowski inequality for weighted dual quermassintegrals

Xiaojuan Chen , Shengyu Tang , Sinan Wang This is my paper

Pith reviewed 2026-07-02 02:14 UTC · model grok-4.3

classification 🧮 math.MG
keywords Brunn-Minkowski inequalitydual quermassintegralsweighted measuresL_p inequalitiesradial weightslog-concavityconvex geometry
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The pith

Under a log-concavity condition on the weight, the L_p Brunn-Minkowski inequality for weighted dual quermassintegrals holds with concavity exponent 1/q.

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

The paper proves an L_p Brunn-Minkowski inequality in weighted spaces for dual quermassintegrals. It shows that when the weight φ satisfies t ↦ log φ(e^t) being concave, the inequality holds with exponent 1/q for p at least 1, better than the usual 1/n. This applies to a class of rotationally invariant measures where the weighted dual quermassintegral integrates |x|^{q-n} φ(|x|). The result generalizes the unweighted case and provides sharper estimates. For smaller p, similar inequalities hold under additional assumptions.

Core claim

We prove that for p ≥ 1 the L_p Brunn-Minkowski inequality holds for the weighted dual quermassintegrals with concavity exponent 1/q, provided that φ is positive and radially non-increasing and t ↦ log φ(e^t) is concave. This improves the exponent 1/n from previous work. The weighted dual quermassintegral is defined by integrating the radial density |x|^{q-n} φ(|x|) for q ∈ (0,n].

What carries the argument

The weighted dual quermassintegral defined by integrating |x|^{q-n} φ(|x|), under the concavity condition on log φ(e^t).

If this is right

  • When φ is identically 1, the result reduces to the classical dual quermassintegral inequality.
  • For p in (0,1), the inequality holds with exponent p/q under more restrictive conditions on the weight.
  • Explicit lower bounds on the range of admissible p are given.

Where Pith is reading between the lines

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

  • The condition t ↦ log φ(e^t) concave aligns with convexity properties used for rotationally invariant measures in probability.
  • This approach might extend to other integral functionals in convex geometry with similar radial weights.
  • The improved exponent could yield sharper volume estimates in weighted convex bodies.

Load-bearing premise

The weight φ must be positive and radially non-increasing with t ↦ log φ(e^t) concave.

What would settle it

A counterexample consisting of a weight φ where t ↦ log φ(e^t) is not concave, for which the L_p Brunn-Minkowski inequality fails to hold with the exponent 1/q.

read the original abstract

We investigate the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals in weighted measure spaces, which is a special class of rotationally invariant measures proposed by Cordero-Erausquin and Rotem [Ann. Probab., {\bf 51} (2023)]. Specifically, the weighted dual quermassintegral is defined by integrating the radial density $|x|^{q-n}\phi(|x|)$ for $q\in(0,n]$, where $\phi$ is a positive radially non-increasing weight, it recovers the classical dual quermassintegral when $\phi\equiv1$. For $p\geq1$, we prove the $L_p$ Brunn-Minkowski inequality with concavity exponent $1/q$ under the condition that $t\mapsto\log\phi(e^t)$ is concave, which is exactly the natural convexity condition from Cordero-Erausquin and Rotem's paper in general, improving the exponent $1/n$. For $p\in(0,1)$, we obtain the result with exponent $p/q$ under more strictly weight assumptions, together with explicit lower bounds for the admissible range of $p$.

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 defines weighted dual quermassintegrals via integration against the radial density |x|^{q-n} φ(|x|) for q ∈ (0,n], where φ > 0 is radially non-increasing. It proves the L_p Brunn-Minkowski inequality for these functionals: for p ≥ 1 the concavity exponent is 1/q whenever t ↦ log φ(e^t) is concave (the Cordero-Erausquin–Rotem condition), improving the classical 1/n; for p ∈ (0,1) the exponent p/q holds under stricter assumptions on φ together with explicit lower bounds on admissible p. The unweighted case φ ≡ 1 is recovered.

Significance. If the derivations hold, the work supplies a direct, natural extension of the L_p Brunn-Minkowski theory to the weighted rotationally invariant measures introduced by Cordero-Erausquin–Rotem, with a strictly better concavity exponent under precisely the convexity condition those authors identified. The explicit range for p < 1 adds concrete information on the boundary of validity. These are load-bearing improvements in the study of geometric inequalities for non-Euclidean measures.

minor comments (2)
  1. [§2] The definition of the weighted dual quermassintegral (presumably in §2) should explicitly record the normalization constant or the precise integral expression to avoid ambiguity when q varies.
  2. [Introduction] A short remark comparing the new exponent 1/q with the classical 1/n for the model case φ ≡ 1 would help readers see the improvement immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment, including the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper takes the concavity condition t ↦ log φ(e^t) directly as an external assumption from the cited Cordero-Erausquin-Rotem work to obtain the improved exponent 1/q; this is presented as an input rather than derived or fitted inside the manuscript. No self-citations appear load-bearing, no quantity is defined in terms of the result it is claimed to produce, and no prediction reduces by construction to a fitted parameter or renamed known pattern. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from convex geometry together with the explicit concavity assumption on the weight function; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption t ↦ log φ(e^t) is concave
    This is the key hypothesis that enables the improved exponent 1/q and is taken from the cited prior work.

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